Theorem karden 3551

Description: If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 3638). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 3550 justify the definition of kard.

Hypotheses

Ref	Expression
karden.1	⊢ A ∈ V
karden.2	⊢ B ∈ V
karden.3	⊢ C = {x∣(x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y)))}
karden.4	⊢ D = {x∣(x ≈ B ∧ ∀y(y ≈ B → (rank ‘x) ⊆ (rank ‘y)))}

Assertion

Ref	Expression
karden	⊢ (C = D ↔ A ≈ B)

Distinct variable group(s):   x,y,A   x,B,y

Proof of Theorem karden

Step	Hyp	Ref	Expression
1		karden.1	. . . . . . . 8 ⊢ A ∈ V
2	1	enref 3295	. . . . . . 7 ⊢ A ≈ A
3		breq1 2065	. . . . . . . 8 ⊢ (w = A → (w ≈ A ↔ A ≈ A))
4	1, 3	cla4ev 1401	. . . . . . 7 ⊢ (A ≈ A → ∃w w ≈ A)
5	2, 4	ax-mp 6	. . . . . 6 ⊢ ∃w w ≈ A
6		abn0 1715	. . . . . 6 ⊢ (¬ {w∣w ≈ A} = ∅ ↔ ∃w w ≈ A)
7	5, 6	mpbir 165	. . . . 5 ⊢ ¬ {w∣w ≈ A} = ∅
8		scott0 3542	. . . . 5 ⊢ ({w∣w ≈ A} = ∅ ↔ {z ∈ {w∣w ≈ A}∣∀y ∈ {w∣w ≈ A} (rank ‘z) ⊆ (rank ‘y)} = ∅)
9	7, 8	mtbi 166	. . . 4 ⊢ ¬ {z ∈ {w∣w ≈ A}∣∀y ∈ {w∣w ≈ A} (rank ‘z) ⊆ (rank ‘y)} = ∅
10		rabn0 1716	. . . 4 ⊢ (¬ {z ∈ {w∣w ≈ A}∣∀y ∈ {w∣w ≈ A} (rank ‘z) ⊆ (rank ‘y)} = ∅ ↔ ∃z ∈ {w∣w ≈ A}∀y ∈ {w∣w ≈ A} (rank ‘z) ⊆ (rank ‘y))
11	9, 10	mpbi 164	. . 3 ⊢ ∃z ∈ {w∣w ≈ A}∀y ∈ {w∣w ≈ A} (rank ‘z) ⊆ (rank ‘y)
12		pm3.26 256	. . . . . . . . 9 ⊢ ((z ≈ A ∧ ∀y(y ≈ A → (rank ‘z) ⊆ (rank ‘y))) → z ≈ A)
13	12	a1i 7	. . . . . . . 8 ⊢ (C = D → ((z ≈ A ∧ ∀y(y ≈ A → (rank ‘z) ⊆ (rank ‘y))) → z ≈ A))
14		karden.3	. . . . . . . . . . . 12 ⊢ C = {x∣(x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y)))}
15		karden.4	. . . . . . . . . . . 12 ⊢ D = {x∣(x ≈ B ∧ ∀y(y ≈ B → (rank ‘x) ⊆ (rank ‘y)))}
16	14, 15	cleq12i 1114	. . . . . . . . . . 11 ⊢ (C = D ↔ {x∣(x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y)))} = {x∣(x ≈ B ∧ ∀y(y ≈ B → (rank ‘x) ⊆ (rank ‘y)))})
17		cleq2ab 1179	. . . . . . . . . . 11 ⊢ ({x∣(x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y)))} = {x∣(x ≈ B ∧ ∀y(y ≈ B → (rank ‘x) ⊆ (rank ‘y)))} ↔ ∀x((x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y))) ↔ (x ≈ B ∧ ∀y(y ≈ B → (rank ‘x) ⊆ (rank ‘y)))))
18	16, 17	bitr 151	. . . . . . . . . 10 ⊢ (C = D ↔ ∀x((x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y))) ↔ (x ≈ B ∧ ∀y(y ≈ B → (rank ‘x) ⊆ (rank ‘y)))))
19		breq1 2065	. . . . . . . . . . . . 13 ⊢ (x = z → (x ≈ A ↔ z ≈ A))
20		fveq2 2832	. . . . . . . . . . . . . . . 16 ⊢ (x = z → (rank ‘x) = (rank ‘z))
21	20	sseq1d 1527	. . . . . . . . . . . . . . 15 ⊢ (x = z → ((rank ‘x) ⊆ (rank ‘y) ↔ (rank ‘z) ⊆ (rank ‘y)))
22	21	imbi2d 464	. . . . . . . . . . . . . 14 ⊢ (x = z → ((y ≈ A → (rank ‘x) ⊆ (rank ‘y)) ↔ (y ≈ A → (rank ‘z) ⊆ (rank ‘y))))
23	22	bialdv 935	. . . . . . . . . . . . 13 ⊢ (x = z → (∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y)) ↔ ∀y(y ≈ A → (rank ‘z) ⊆ (rank ‘y))))
24	19, 23	anbi12d 476	. . . . . . . . . . . 12 ⊢ (x = z → ((x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y))) ↔ (z ≈ A ∧ ∀y(y ≈ A → (rank ‘z) ⊆ (rank ‘y)))))
25		breq1 2065	. . . . . . . . . . . . 13 ⊢ (x = z → (x ≈ B ↔ z ≈ B))
26	21	imbi2d 464	. . . . . . . . . . . . . 14 ⊢ (x = z → ((y ≈ B → (rank ‘x) ⊆ (rank ‘y)) ↔ (y ≈ B → (rank ‘z) ⊆ (rank ‘y))))
27	26	bialdv 935	. . . . . . . . . . . . 13 ⊢ (x = z → (∀y(y ≈ B → (rank ‘x) ⊆ (rank ‘y)) ↔ ∀y(y ≈ B → (rank ‘z) ⊆ (rank ‘y))))
28	25, 27	anbi12d 476	. . . . . . . . . . . 12 ⊢ (x = z → ((x ≈ B ∧ ∀y(y ≈ B → (rank ‘x) ⊆ (rank ‘y))) ↔ (z ≈ B ∧ ∀y(y ≈ B → (rank ‘z) ⊆ (rank ‘y)))))
29	24, 28	bibi12d 477	. . . . . . . . . . 11 ⊢ (x = z → (((x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y))) ↔ (x ≈ B ∧ ∀y(y ≈ B → (rank ‘x) ⊆ (rank ‘y)))) ↔ ((z ≈ A ∧ ∀y(y ≈ A → (rank ‘z) ⊆ (rank ‘y))) ↔ (z ≈ B ∧ ∀y(y ≈ B → (rank ‘z) ⊆ (rank ‘y))))))
30	29	a4b1 928	. . . . . . . . . 10 ⊢ (∀x((x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y))) ↔ (x ≈ B ∧ ∀y(y ≈ B → (rank ‘x) ⊆ (rank ‘y)))) → ((z ≈ A ∧ ∀y(y ≈ A → (rank ‘z) ⊆ (rank ‘y))) ↔ (z ≈ B ∧ ∀y(y ≈ B → (rank ‘z) ⊆ (rank ‘y)))))
31	18, 30	sylbi 174	. . . . . . . . 9 ⊢ (C = D → ((z ≈ A ∧ ∀y(y ≈ A → (rank ‘z) ⊆ (rank ‘y))) ↔ (z ≈ B ∧ ∀y(y ≈ B → (rank ‘z) ⊆ (rank ‘y)))))
32		pm3.26 256	. . . . . . . . 9 ⊢ ((z ≈ B ∧ ∀y(y ≈ B → (rank ‘z) ⊆ (rank ‘y))) → z ≈ B)
33	31, 32	syl6bi 187	. . . . . . . 8 ⊢ (C = D → ((z ≈ A ∧ ∀y(y ≈ A → (rank ‘z) ⊆ (rank ‘y))) → z ≈ B))
34	13, 33	jcad 455	. . . . . . 7 ⊢ (C = D → ((z ≈ A ∧ ∀y(y ≈ A → (rank ‘z) ⊆ (rank ‘y))) → (z ≈ A ∧ z ≈ B)))
35		entrt 3319	. . . . . . . 8 ⊢ ((A ≈ z ∧ z ≈ B) → A ≈ B)
36	1	ensym 3317	. . . . . . . 8 ⊢ (z ≈ A → A ≈ z)
37	35, 36	sylan 343	. . . . . . 7 ⊢ ((z ≈ A ∧ z ≈ B) → A ≈ B)
38	34, 37	syl6 23	. . . . . 6 ⊢ (C = D → ((z ≈ A ∧ ∀y(y ≈ A → (rank ‘z) ⊆ (rank ‘y))) → A ≈ B))
39		visset 1350	. . . . . . . 8 ⊢ z ∈ V
40		breq1 2065	. . . . . . . 8 ⊢ (w = z → (w ≈ A ↔ z ≈ A))
41	39, 40	elab 1415	. . . . . . 7 ⊢ (z ∈ {w∣w ≈ A} ↔ z ≈ A)
42		df-ral 1205	. . . . . . . 8 ⊢ (∀y ∈ {w∣w ≈ A} (rank ‘z) ⊆ (rank ‘y) ↔ ∀y(y ∈ {w∣w ≈ A} → (rank ‘z) ⊆ (rank ‘y)))
43		visset 1350	. . . . . . . . . . 11 ⊢ y ∈ V
44		breq1 2065	. . . . . . . . . . 11 ⊢ (w = y → (w ≈ A ↔ y ≈ A))
45	43, 44	elab 1415	. . . . . . . . . 10 ⊢ (y ∈ {w∣w ≈ A} ↔ y ≈ A)
46	45	imbi1i 161	. . . . . . . . 9 ⊢ ((y ∈ {w∣w ≈ A} → (rank ‘z) ⊆ (rank ‘y)) ↔ (y ≈ A → (rank ‘z) ⊆ (rank ‘y)))
47	46	bial 695	. . . . . . . 8 ⊢ (∀y(y ∈ {w∣w ≈ A} → (rank ‘z) ⊆ (rank ‘y)) ↔ ∀y(y ≈ A → (rank ‘z) ⊆ (rank ‘y)))
48	42, 47	bitr 151	. . . . . . 7 ⊢ (∀y ∈ {w∣w ≈ A} (rank ‘z) ⊆ (rank ‘y) ↔ ∀y(y ≈ A → (rank ‘z) ⊆ (rank ‘y)))
49	41, 48	anbi12i 369	. . . . . 6 ⊢ ((z ∈ {w∣w ≈ A} ∧ ∀y ∈ {w∣w ≈ A} (rank ‘z) ⊆ (rank ‘y)) ↔ (z ≈ A ∧ ∀y(y ≈ A → (rank ‘z) ⊆ (rank ‘y))))
50	38, 49	syl5ib 181	. . . . 5 ⊢ (C = D → ((z ∈ {w∣w ≈ A} ∧ ∀y ∈ {w∣w ≈ A} (rank ‘z) ⊆ (rank ‘y)) → A ≈ B))
51	50	exp3a 292	. . . 4 ⊢ (C = D → (z ∈ {w∣w ≈ A} → (∀y ∈ {w∣w ≈ A} (rank ‘z) ⊆ (rank ‘y) → A ≈ B)))
52	51	r19.23adv 1286	. . 3 ⊢ (C = D → (∃z ∈ {w∣w ≈ A}∀y ∈ {w∣w ≈ A} (rank ‘z) ⊆ (rank ‘y) → A ≈ B))
53	11, 52	mpi 44	. 2 ⊢ (C = D → A ≈ B)
54		karden.2	. . . . . 6 ⊢ B ∈ V
55		enen2 3376	. . . . . 6 ⊢ ((B ∈ V ∧ A ≈ B) → (x ≈ A ↔ x ≈ B))
56	54, 55	mpan 518	. . . . 5 ⊢ (A ≈ B → (x ≈ A ↔ x ≈ B))
57		enen2 3376	. . . . . . . 8 ⊢ ((B ∈ V ∧ A ≈ B) → (y ≈ A ↔ y ≈ B))
58	54, 57	mpan 518	. . . . . . 7 ⊢ (A ≈ B → (y ≈ A ↔ y ≈ B))
59	58	imbi1d 465	. . . . . 6 ⊢ (A ≈ B → ((y ≈ A → (rank ‘x) ⊆ (rank ‘y)) ↔ (y ≈ B → (rank ‘x) ⊆ (rank ‘y))))
60	59	bialdv 935	. . . . 5 ⊢ (A ≈ B → (∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y)) ↔ ∀y(y ≈ B → (rank ‘x) ⊆ (rank ‘y))))
61	56, 60	anbi12d 476	. . . 4 ⊢ (A ≈ B → ((x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y))) ↔ (x ≈ B ∧ ∀y(y ≈ B → (rank ‘x) ⊆ (rank ‘y)))))
62	61	biabdv 1183	. . 3 ⊢ (A ≈ B → {x∣(x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y)))} = {x∣(x ≈ B ∧ ∀y(y ≈ B → (rank ‘x) ⊆ (rank ‘y)))})
63	62, 14, 15	3eqtr4g 1147	. 2 ⊢ (A ≈ B → C = D)
64	53, 63	impbi 139	1 ⊢ (C = D ↔ A ≈ B)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  {crab 1204  Vcvv 1348   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054   ‘cfv 2422   ≈ cen 3271  rankcrnk 3486

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-iin 1997  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-er 3200  df-en 3274  df-r1 3487  df-rank 3488

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