Theorem scott0 3542

Description: Scott's trick collects all sets that have a certain property and are of smallest possible rank. This theorem shows that the resulting collection, expressed as in Equation 9.3 of [Jech] p. 72, contains at least one representative with the property, if there is one. In other words, the collection is empty iff no set has the property (i.e. A is empty).

Assertion

Ref	Expression
scott0	⊢ (A = ∅ ↔ {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} = ∅)

Distinct variable group(s):   x,y,A

Proof of Theorem scott0

Step	Hyp	Ref	Expression
1		rabeq 1346	. . 3 ⊢ (A = ∅ → {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} = {x ∈ ∅∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)})
2		rab0 1718	. . 3 ⊢ {x ∈ ∅∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} = ∅
3	1, 2	syl6eq 1140	. 2 ⊢ (A = ∅ → {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} = ∅)
4		n0 1714	. . . . . . . . 9 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
5		hbre1 1239	. . . . . . . . . 10 ⊢ (∃x ∈ A (rank ‘x) = (rank ‘x) → ∀x∃x ∈ A (rank ‘x) = (rank ‘x))
6		cleqid 1102	. . . . . . . . . . 11 ⊢ (rank ‘x) = (rank ‘x)
7		ra4e 1244	. . . . . . . . . . 11 ⊢ ((x ∈ A ∧ (rank ‘x) = (rank ‘x)) → ∃x ∈ A (rank ‘x) = (rank ‘x))
8	6, 7	mpan2 519	. . . . . . . . . 10 ⊢ (x ∈ A → ∃x ∈ A (rank ‘x) = (rank ‘x))
9	5, 8	19.23ai 746	. . . . . . . . 9 ⊢ (∃x x ∈ A → ∃x ∈ A (rank ‘x) = (rank ‘x))
10	4, 9	sylbi 174	. . . . . . . 8 ⊢ (¬ A = ∅ → ∃x ∈ A (rank ‘x) = (rank ‘x))
11		fvex 2838	. . . . . . . . . . . 12 ⊢ (rank ‘x) ∈ V
12		cleq1 1107	. . . . . . . . . . . . 13 ⊢ (y = (rank ‘x) → (y = (rank ‘x) ↔ (rank ‘x) = (rank ‘x)))
13	12	anbi2d 468	. . . . . . . . . . . 12 ⊢ (y = (rank ‘x) → ((x ∈ A ∧ y = (rank ‘x)) ↔ (x ∈ A ∧ (rank ‘x) = (rank ‘x))))
14	11, 13	cla4ev 1401	. . . . . . . . . . 11 ⊢ ((x ∈ A ∧ (rank ‘x) = (rank ‘x)) → ∃y(x ∈ A ∧ y = (rank ‘x)))
15	14	19.22i 723	. . . . . . . . . 10 ⊢ (∃x(x ∈ A ∧ (rank ‘x) = (rank ‘x)) → ∃x∃y(x ∈ A ∧ y = (rank ‘x)))
16		excom 728	. . . . . . . . . 10 ⊢ (∃y∃x(x ∈ A ∧ y = (rank ‘x)) ↔ ∃x∃y(x ∈ A ∧ y = (rank ‘x)))
17	15, 16	sylibr 175	. . . . . . . . 9 ⊢ (∃x(x ∈ A ∧ (rank ‘x) = (rank ‘x)) → ∃y∃x(x ∈ A ∧ y = (rank ‘x)))
18		df-rex 1206	. . . . . . . . 9 ⊢ (∃x ∈ A (rank ‘x) = (rank ‘x) ↔ ∃x(x ∈ A ∧ (rank ‘x) = (rank ‘x)))
19		df-rex 1206	. . . . . . . . . 10 ⊢ (∃x ∈ A y = (rank ‘x) ↔ ∃x(x ∈ A ∧ y = (rank ‘x)))
20	19	biex 733	. . . . . . . . 9 ⊢ (∃y∃x ∈ A y = (rank ‘x) ↔ ∃y∃x(x ∈ A ∧ y = (rank ‘x)))
21	17, 18, 20	3imtr4 192	. . . . . . . 8 ⊢ (∃x ∈ A (rank ‘x) = (rank ‘x) → ∃y∃x ∈ A y = (rank ‘x))
22	10, 21	syl 12	. . . . . . 7 ⊢ (¬ A = ∅ → ∃y∃x ∈ A y = (rank ‘x))
23		abn0 1715	. . . . . . 7 ⊢ (¬ {y∣∃x ∈ A y = (rank ‘x)} = ∅ ↔ ∃y∃x ∈ A y = (rank ‘x))
24	22, 23	sylibr 175	. . . . . 6 ⊢ (¬ A = ∅ → ¬ {y∣∃x ∈ A y = (rank ‘x)} = ∅)
25		hbab1 1095	. . . . . . . . . 10 ⊢ (z ∈ {y∣∃x ∈ A y = (rank ‘x)} → ∀y z ∈ {y∣∃x ∈ A y = (rank ‘x)})
26		ax-17 925	. . . . . . . . . 10 ⊢ (z ∈ On → ∀y z ∈ On)
27	25, 26	dfss2f 1499	. . . . . . . . 9 ⊢ ({y∣∃x ∈ A y = (rank ‘x)} ⊆ On ↔ ∀y(y ∈ {y∣∃x ∈ A y = (rank ‘x)} → y ∈ On))
28		abid 1094	. . . . . . . . . 10 ⊢ (y ∈ {y∣∃x ∈ A y = (rank ‘x)} ↔ ∃x ∈ A y = (rank ‘x))
29		rankon 3515	. . . . . . . . . . . . 13 ⊢ (rank ‘x) ∈ On
30		eleq1 1149	. . . . . . . . . . . . 13 ⊢ (y = (rank ‘x) → (y ∈ On ↔ (rank ‘x) ∈ On))
31	29, 30	mpbiri 169	. . . . . . . . . . . 12 ⊢ (y = (rank ‘x) → y ∈ On)
32	31	a1i 7	. . . . . . . . . . 11 ⊢ (x ∈ A → (y = (rank ‘x) → y ∈ On))
33	32	r19.23aiv 1284	. . . . . . . . . 10 ⊢ (∃x ∈ A y = (rank ‘x) → y ∈ On)
34	28, 33	sylbi 174	. . . . . . . . 9 ⊢ (y ∈ {y∣∃x ∈ A y = (rank ‘x)} → y ∈ On)
35	27, 34	mpgbir 686	. . . . . . . 8 ⊢ {y∣∃x ∈ A y = (rank ‘x)} ⊆ On
36		onint 2261	. . . . . . . 8 ⊢ (({y∣∃x ∈ A y = (rank ‘x)} ⊆ On ∧ ¬ {y∣∃x ∈ A y = (rank ‘x)} = ∅) → ∩{y∣∃x ∈ A y = (rank ‘x)} ∈ {y∣∃x ∈ A y = (rank ‘x)})
37	35, 36	mpan 518	. . . . . . 7 ⊢ (¬ {y∣∃x ∈ A y = (rank ‘x)} = ∅ → ∩{y∣∃x ∈ A y = (rank ‘x)} ∈ {y∣∃x ∈ A y = (rank ‘x)})
38	11	dfiin2 2015	. . . . . . . 8 ⊢ ∩x ∈ A (rank ‘x) = ∩{y∣∃x ∈ A y = (rank ‘x)}
39	38	eleq1i 1152	. . . . . . 7 ⊢ (∩x ∈ A (rank ‘x) ∈ {y∣∃x ∈ A y = (rank ‘x)} ↔ ∩{y∣∃x ∈ A y = (rank ‘x)} ∈ {y∣∃x ∈ A y = (rank ‘x)})
40	37, 39	sylibr 175	. . . . . 6 ⊢ (¬ {y∣∃x ∈ A y = (rank ‘x)} = ∅ → ∩x ∈ A (rank ‘x) ∈ {y∣∃x ∈ A y = (rank ‘x)})
41	24, 40	syl 12	. . . . 5 ⊢ (¬ A = ∅ → ∩x ∈ A (rank ‘x) ∈ {y∣∃x ∈ A y = (rank ‘x)})
42		hbii1 2013	. . . . . . . . 9 ⊢ (y ∈ ∩x ∈ A (rank ‘x) → ∀x y ∈ ∩x ∈ A (rank ‘x))
43	42	hbeleq 1173	. . . . . . . 8 ⊢ (y = ∩x ∈ A (rank ‘x) → ∀x y = ∩x ∈ A (rank ‘x))
44		cleq1 1107	. . . . . . . 8 ⊢ (y = ∩x ∈ A (rank ‘x) → (y = (rank ‘x) ↔ ∩x ∈ A (rank ‘x) = (rank ‘x)))
45	43, 44	birexd 1218	. . . . . . 7 ⊢ (y = ∩x ∈ A (rank ‘x) → (∃x ∈ A y = (rank ‘x) ↔ ∃x ∈ A ∩x ∈ A (rank ‘x) = (rank ‘x)))
46	45	elabg 1417	. . . . . 6 ⊢ (∩x ∈ A (rank ‘x) ∈ {y∣∃x ∈ A y = (rank ‘x)} → (∩x ∈ A (rank ‘x) ∈ {y∣∃x ∈ A y = (rank ‘x)} ↔ ∃x ∈ A ∩x ∈ A (rank ‘x) = (rank ‘x)))
47	46	ibi 449	. . . . 5 ⊢ (∩x ∈ A (rank ‘x) ∈ {y∣∃x ∈ A y = (rank ‘x)} → ∃x ∈ A ∩x ∈ A (rank ‘x) = (rank ‘x))
48		sseq1 1521	. . . . . . . 8 ⊢ (∩x ∈ A (rank ‘x) = (rank ‘x) → (∩x ∈ A (rank ‘x) ⊆ (rank ‘y) ↔ (rank ‘x) ⊆ (rank ‘y)))
49		ssid 1519	. . . . . . . . . 10 ⊢ (rank ‘y) ⊆ (rank ‘y)
50		fveq2 2832	. . . . . . . . . . . 12 ⊢ (x = y → (rank ‘x) = (rank ‘y))
51	50	sseq1d 1527	. . . . . . . . . . 11 ⊢ (x = y → ((rank ‘x) ⊆ (rank ‘y) ↔ (rank ‘y) ⊆ (rank ‘y)))
52	51	rcla4ev 1403	. . . . . . . . . 10 ⊢ ((y ∈ A ∧ (rank ‘y) ⊆ (rank ‘y)) → ∃x ∈ A (rank ‘x) ⊆ (rank ‘y))
53	49, 52	mpan2 519	. . . . . . . . 9 ⊢ (y ∈ A → ∃x ∈ A (rank ‘x) ⊆ (rank ‘y))
54		iinss 2025	. . . . . . . . 9 ⊢ (∃x ∈ A (rank ‘x) ⊆ (rank ‘y) → ∩x ∈ A (rank ‘x) ⊆ (rank ‘y))
55	53, 54	syl 12	. . . . . . . 8 ⊢ (y ∈ A → ∩x ∈ A (rank ‘x) ⊆ (rank ‘y))
56	48, 55	syl5bi 183	. . . . . . 7 ⊢ (∩x ∈ A (rank ‘x) = (rank ‘x) → (y ∈ A → (rank ‘x) ⊆ (rank ‘y)))
57	56	r19.21aiv 1259	. . . . . 6 ⊢ (∩x ∈ A (rank ‘x) = (rank ‘x) → ∀y ∈ A (rank ‘x) ⊆ (rank ‘y))
58	57	r19.22si 1275	. . . . 5 ⊢ (∃x ∈ A ∩x ∈ A (rank ‘x) = (rank ‘x) → ∃x ∈ A ∀y ∈ A (rank ‘x) ⊆ (rank ‘y))
59	41, 47, 58	3syl 21	. . . 4 ⊢ (¬ A = ∅ → ∃x ∈ A ∀y ∈ A (rank ‘x) ⊆ (rank ‘y))
60		rabn0 1716	. . . 4 ⊢ (¬ {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} = ∅ ↔ ∃x ∈ A ∀y ∈ A (rank ‘x) ⊆ (rank ‘y))
61	59, 60	sylibr 175	. . 3 ⊢ (¬ A = ∅ → ¬ {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} = ∅)
62	61	a3i 69	. 2 ⊢ ({x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} = ∅ → A = ∅)
63	3, 62	impbi 139	1 ⊢ (A = ∅ ↔ {x ∈ A∣∀y ∈ A (rank ‘x) ⊆ (rank ‘y)} = ∅)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  {crab 1204   ⊆ wss 1487  ∅c0 1707  ∩cint 1965  ∩ciin 1995  Oncon0 2199   ‘cfv 2422  rankcrnk 3486

This theorem is referenced by:  scott0s 3544  cplem1 3545  karden 3551

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-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

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