Theorem phplem5 3407

Description: Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers.

Hypotheses

Ref	Expression
phplem3.1	⊢ A ∈ V
phplem3.2	⊢ B ∈ V

Assertion

Ref	Expression
phplem5	⊢ ((A ∈ ω ∧ B ∈ ω) → (suc A ≈ suc B → A ≈ B))

Proof of Theorem phplem5

Step	Hyp	Ref	Expression
1		entrt 3319	. . . . . 6 ⊢ ((A ≈ (suc B ∖ {(f ‘A)}) ∧ (suc B ∖ {(f ‘A)}) ≈ B) → A ≈ B)
2		f1of1 2799	. . . . . . . . . 10 ⊢ (f:suc A–1-1-onto→suc B → f:suc A–1-1→suc B)
3		sssucid 2300	. . . . . . . . . 10 ⊢ A ⊆ suc A
4	2, 3	jctir 241	. . . . . . . . 9 ⊢ (f:suc A–1-1-onto→suc B → (f:suc A–1-1→suc B ∧ A ⊆ suc A))
5		f1ores 2813	. . . . . . . . 9 ⊢ ((f:suc A–1-1→suc B ∧ A ⊆ suc A) → (f ↾ A):A–1-1-onto→(f “ A))
6		phplem3.1	. . . . . . . . . 10 ⊢ A ∈ V
7	6	f1oen 3301	. . . . . . . . 9 ⊢ ((f ↾ A):A–1-1-onto→(f “ A) → A ≈ (f “ A))
8	4, 5, 7	3syl 21	. . . . . . . 8 ⊢ (f:suc A–1-1-onto→suc B → A ≈ (f “ A))
9	8	adantl 305	. . . . . . 7 ⊢ ((A ∈ ω ∧ f:suc A–1-1-onto→suc B) → A ≈ (f “ A))
10		nnord 2381	. . . . . . . . 9 ⊢ (A ∈ ω → Ord A)
11		orddif 2326	. . . . . . . . 9 ⊢ (Ord A → A = (suc A ∖ {A}))
12		imaeq2 2603	. . . . . . . . 9 ⊢ (A = (suc A ∖ {A}) → (f “ A) = (f “ (suc A ∖ {A})))
13	10, 11, 12	3syl 21	. . . . . . . 8 ⊢ (A ∈ ω → (f “ A) = (f “ (suc A ∖ {A})))
14		f1ofn 2801	. . . . . . . . . 10 ⊢ (f:suc A–1-1-onto→suc B → f Fn suc A)
15	6	sucid 2304	. . . . . . . . . . 11 ⊢ A ∈ suc A
16		fnsnfv 2861	. . . . . . . . . . 11 ⊢ ((f Fn suc A ∧ A ∈ suc A) → {(f ‘A)} = (f “ {A}))
17	15, 16	mpan2 519	. . . . . . . . . 10 ⊢ (f Fn suc A → {(f ‘A)} = (f “ {A}))
18		difeq2 1583	. . . . . . . . . 10 ⊢ ({(f ‘A)} = (f “ {A}) → ((f “ suc A) ∖ {(f ‘A)}) = ((f “ suc A) ∖ (f “ {A})))
19	14, 17, 18	3syl 21	. . . . . . . . 9 ⊢ (f:suc A–1-1-onto→suc B → ((f “ suc A) ∖ {(f ‘A)}) = ((f “ suc A) ∖ (f “ {A})))
20		imadmrn 2610	. . . . . . . . . . . . 13 ⊢ (f “ dom f) = ran f
21	20	cleqcomi 1105	. . . . . . . . . . . 12 ⊢ ran f = (f “ dom f)
22	21	a1i 7	. . . . . . . . . . 11 ⊢ (f:suc A–1-1-onto→suc B → ran f = (f “ dom f))
23		f1ofo 2806	. . . . . . . . . . . 12 ⊢ (f:suc A–1-1-onto→suc B → f:suc A–onto→suc B)
24		forn 2789	. . . . . . . . . . . 12 ⊢ (f:suc A–onto→suc B → ran f = suc B)
25	23, 24	syl 12	. . . . . . . . . . 11 ⊢ (f:suc A–1-1-onto→suc B → ran f = suc B)
26		fndm 2723	. . . . . . . . . . . 12 ⊢ (f Fn suc A → dom f = suc A)
27		imaeq2 2603	. . . . . . . . . . . 12 ⊢ (dom f = suc A → (f “ dom f) = (f “ suc A))
28	14, 26, 27	3syl 21	. . . . . . . . . . 11 ⊢ (f:suc A–1-1-onto→suc B → (f “ dom f) = (f “ suc A))
29	22, 25, 28	3eqtr3d 1133	. . . . . . . . . 10 ⊢ (f:suc A–1-1-onto→suc B → suc B = (f “ suc A))
30	29	difeq1d 1587	. . . . . . . . 9 ⊢ (f:suc A–1-1-onto→suc B → (suc B ∖ {(f ‘A)}) = ((f “ suc A) ∖ {(f ‘A)}))
31		f1o3 2805	. . . . . . . . . . 11 ⊢ (f:suc A–1-1-onto→suc B ↔ (f:suc A–onto→suc B ∧ Fun ◡f))
32	31	pm3.27bd 263	. . . . . . . . . 10 ⊢ (f:suc A–1-1-onto→suc B → Fun ◡f)
33		imadif 2714	. . . . . . . . . 10 ⊢ (Fun ◡f → (f “ (suc A ∖ {A})) = ((f “ suc A) ∖ (f “ {A})))
34	32, 33	syl 12	. . . . . . . . 9 ⊢ (f:suc A–1-1-onto→suc B → (f “ (suc A ∖ {A})) = ((f “ suc A) ∖ (f “ {A})))
35	19, 30, 34	3eqtr4rd 1135	. . . . . . . 8 ⊢ (f:suc A–1-1-onto→suc B → (f “ (suc A ∖ {A})) = (suc B ∖ {(f ‘A)}))
36	13, 35	sylan9eq 1144	. . . . . . 7 ⊢ ((A ∈ ω ∧ f:suc A–1-1-onto→suc B) → (f “ A) = (suc B ∖ {(f ‘A)}))
37	9, 36	breqtrd 2081	. . . . . 6 ⊢ ((A ∈ ω ∧ f:suc A–1-1-onto→suc B) → A ≈ (suc B ∖ {(f ‘A)}))
38		phplem3.2	. . . . . . . . 9 ⊢ B ∈ V
39		fvex 2838	. . . . . . . . 9 ⊢ (f ‘A) ∈ V
40	38, 39	phplem4 3406	. . . . . . . 8 ⊢ ((B ∈ ω ∧ (f ‘A) ∈ suc B) → B ≈ (suc B ∖ {(f ‘A)}))
41		fnfvrn 2889	. . . . . . . . . . 11 ⊢ ((f Fn suc A ∧ A ∈ suc A) → (f ‘A) ∈ ran f)
42	15, 41	mpan2 519	. . . . . . . . . 10 ⊢ (f Fn suc A → (f ‘A) ∈ ran f)
43	14, 42	syl 12	. . . . . . . . 9 ⊢ (f:suc A–1-1-onto→suc B → (f ‘A) ∈ ran f)
44	24	eleq2d 1156	. . . . . . . . . 10 ⊢ (f:suc A–onto→suc B → ((f ‘A) ∈ ran f ↔ (f ‘A) ∈ suc B))
45	23, 44	syl 12	. . . . . . . . 9 ⊢ (f:suc A–1-1-onto→suc B → ((f ‘A) ∈ ran f ↔ (f ‘A) ∈ suc B))
46	43, 45	mpbid 170	. . . . . . . 8 ⊢ (f:suc A–1-1-onto→suc B → (f ‘A) ∈ suc B)
47	40, 46	sylan2 346	. . . . . . 7 ⊢ ((B ∈ ω ∧ f:suc A–1-1-onto→suc B) → B ≈ (suc B ∖ {(f ‘A)}))
48	38	sucex 2303	. . . . . . . . 9 ⊢ suc B ∈ V
49		difss 1596	. . . . . . . . 9 ⊢ (suc B ∖ {(f ‘A)}) ⊆ suc B
50	48, 49	ssexi 1701	. . . . . . . 8 ⊢ (suc B ∖ {(f ‘A)}) ∈ V
51		ener 3313	. . . . . . . 8 ⊢ Er ≈
52	38, 50, 51	ersym 3209	. . . . . . 7 ⊢ (B ≈ (suc B ∖ {(f ‘A)}) → (suc B ∖ {(f ‘A)}) ≈ B)
53	47, 52	syl 12	. . . . . 6 ⊢ ((B ∈ ω ∧ f:suc A–1-1-onto→suc B) → (suc B ∖ {(f ‘A)}) ≈ B)
54	1, 37, 53	syl2an 349	. . . . 5 ⊢ (((A ∈ ω ∧ f:suc A–1-1-onto→suc B) ∧ (B ∈ ω ∧ f:suc A–1-1-onto→suc B)) → A ≈ B)
55	54	anandirs 395	. . . 4 ⊢ (((A ∈ ω ∧ B ∈ ω) ∧ f:suc A–1-1-onto→suc B) → A ≈ B)
56	55	exp 291	. . 3 ⊢ ((A ∈ ω ∧ B ∈ ω) → (f:suc A–1-1-onto→suc B → A ≈ B))
57	56	19.23adv 954	. 2 ⊢ ((A ∈ ω ∧ B ∈ ω) → (∃f f:suc A–1-1-onto→suc B → A ≈ B))
58	48	bren 3282	. 2 ⊢ (suc A ≈ suc B ↔ ∃f f:suc A–1-1-onto→suc B)
59	57, 58	syl5ib 181	1 ⊢ ((A ∈ ω ∧ B ∈ ω) → (suc A ≈ suc B → A ≈ B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484   ⊆ wss 1487  {csn 1808   class class class wbr 2054  Ord word 2198  suc csuc 2201  ωcom 2372  ^◡ccnv 2409  dom cdm 2410  ran crn 2411   ↾ cres 2412   “ cima 2413  Fun wfun 2416   Fn wfn 2417  –1-1→wf1 2419  –onto→wfo 2420  –1-1-onto→wf1o 2421   ‘cfv 2422   ≈ cen 3271

This theorem is referenced by:  nneneq 3408

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

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  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-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  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-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-er 3200  df-en 3274

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