Theorem infensuc 8138

Description: Any infinite ordinal is equinumerous to its successor. Exercise 7 of [TakeutiZaring] p. 88. Proved without the Axiom of Infinity. (Contributed by NM, 30-Oct-2003.) (Revised by Mario Carneiro, 13-Jan-2013.)

Assertion

Ref	Expression
infensuc	⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ≈ suc 𝐴)

Proof of Theorem infensuc

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onprc 6984	. . . . 5 ⊢ ¬ On ∈ V
2		eleq1 2689	. . . . 5 ⊢ (ω = On → (ω ∈ V ↔ On ∈ V))
3	1, 2	mtbiri 317	. . . 4 ⊢ (ω = On → ¬ ω ∈ V)
4		ssexg 4804	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ 𝐴 ∈ On) → ω ∈ V)
5	4	ancoms 469	. . . 4 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ω ∈ V)
6	3, 5	nsyl3 133	. . 3 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ¬ ω = On)
7		omon 7076	. . . 4 ⊢ (ω ∈ On ∨ ω = On)
8	7	ori 390	. . 3 ⊢ (¬ ω ∈ On → ω = On)
9	6, 8	nsyl2 142	. 2 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ω ∈ On)
10		id 22	. . . . . . 7 ⊢ (𝑥 = ω → 𝑥 = ω)
11		suceq 5790	. . . . . . 7 ⊢ (𝑥 = ω → suc 𝑥 = suc ω)
12	10, 11	breq12d 4666	. . . . . 6 ⊢ (𝑥 = ω → (𝑥 ≈ suc 𝑥 ↔ ω ≈ suc ω))
13		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
14		suceq 5790	. . . . . . 7 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
15	13, 14	breq12d 4666	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑥 ↔ 𝑦 ≈ suc 𝑦))
16		id 22	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
17		suceq 5790	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
18	16, 17	breq12d 4666	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ suc 𝑥 ↔ suc 𝑦 ≈ suc suc 𝑦))
19		id 22	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
20		suceq 5790	. . . . . . 7 ⊢ (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
21	19, 20	breq12d 4666	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ suc 𝑥 ↔ 𝐴 ≈ suc 𝐴))
22		limom 7080	. . . . . . 7 ⊢ Lim ω
23	22	limensuci 8136	. . . . . 6 ⊢ (ω ∈ On → ω ≈ suc ω)
24		vex 3203	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
25	24	sucex 7011	. . . . . . . . . 10 ⊢ suc 𝑦 ∈ V
26		en2sn 8037	. . . . . . . . . 10 ⊢ ((𝑦 ∈ V ∧ suc 𝑦 ∈ V) → {𝑦} ≈ {suc 𝑦})
27	24, 25, 26	mp2an 708	. . . . . . . . 9 ⊢ {𝑦} ≈ {suc 𝑦}
28		eloni 5733	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ On → Ord 𝑦)
29		ordirr 5741	. . . . . . . . . . . . 13 ⊢ (Ord 𝑦 → ¬ 𝑦 ∈ 𝑦)
30	28, 29	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → ¬ 𝑦 ∈ 𝑦)
31		disjsn 4246	. . . . . . . . . . . 12 ⊢ ((𝑦 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑦)
32	30, 31	sylibr 224	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (𝑦 ∩ {𝑦}) = ∅)
33		eloni 5733	. . . . . . . . . . . . 13 ⊢ (suc 𝑦 ∈ On → Ord suc 𝑦)
34		ordirr 5741	. . . . . . . . . . . . 13 ⊢ (Ord suc 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
35	33, 34	syl 17	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ∈ On → ¬ suc 𝑦 ∈ suc 𝑦)
36		sucelon 7017	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On ↔ suc 𝑦 ∈ On)
37		disjsn 4246	. . . . . . . . . . . 12 ⊢ ((suc 𝑦 ∩ {suc 𝑦}) = ∅ ↔ ¬ suc 𝑦 ∈ suc 𝑦)
38	35, 36, 37	3imtr4i 281	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (suc 𝑦 ∩ {suc 𝑦}) = ∅)
39	32, 38	jca 554	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅))
40		unen 8040	. . . . . . . . . . . 12 ⊢ (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (suc 𝑦 ∪ {suc 𝑦}))
41		df-suc 5729	. . . . . . . . . . . 12 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
42		df-suc 5729	. . . . . . . . . . . 12 ⊢ suc suc 𝑦 = (suc 𝑦 ∪ {suc 𝑦})
43	40, 41, 42	3brtr4g 4687	. . . . . . . . . . 11 ⊢ (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → suc 𝑦 ≈ suc suc 𝑦)
44	43	ex 450	. . . . . . . . . 10 ⊢ ((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) → (((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅) → suc 𝑦 ≈ suc suc 𝑦))
45	39, 44	syl5 34	. . . . . . . . 9 ⊢ ((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) → (𝑦 ∈ On → suc 𝑦 ≈ suc suc 𝑦))
46	27, 45	mpan2 707	. . . . . . . 8 ⊢ (𝑦 ≈ suc 𝑦 → (𝑦 ∈ On → suc 𝑦 ≈ suc suc 𝑦))
47	46	com12 32	. . . . . . 7 ⊢ (𝑦 ∈ On → (𝑦 ≈ suc 𝑦 → suc 𝑦 ≈ suc suc 𝑦))
48	47	ad2antrr 762	. . . . . 6 ⊢ (((𝑦 ∈ On ∧ ω ∈ On) ∧ ω ⊆ 𝑦) → (𝑦 ≈ suc 𝑦 → suc 𝑦 ≈ suc suc 𝑦))
49		vex 3203	. . . . . . . . 9 ⊢ 𝑥 ∈ V
50		limensuc 8137	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ≈ suc 𝑥)
51	49, 50	mpan 706	. . . . . . . 8 ⊢ (Lim 𝑥 → 𝑥 ≈ suc 𝑥)
52	51	ad2antrr 762	. . . . . . 7 ⊢ (((Lim 𝑥 ∧ ω ∈ On) ∧ ω ⊆ 𝑥) → 𝑥 ≈ suc 𝑥)
53	52	a1d 25	. . . . . 6 ⊢ (((Lim 𝑥 ∧ ω ∈ On) ∧ ω ⊆ 𝑥) → (∀𝑦 ∈ 𝑥 (ω ⊆ 𝑦 → 𝑦 ≈ suc 𝑦) → 𝑥 ≈ suc 𝑥))
54	12, 15, 18, 21, 23, 48, 53	tfindsg 7060	. . . . 5 ⊢ (((𝐴 ∈ On ∧ ω ∈ On) ∧ ω ⊆ 𝐴) → 𝐴 ≈ suc 𝐴)
55	54	exp31 630	. . . 4 ⊢ (𝐴 ∈ On → (ω ∈ On → (ω ⊆ 𝐴 → 𝐴 ≈ suc 𝐴)))
56	55	com23 86	. . 3 ⊢ (𝐴 ∈ On → (ω ⊆ 𝐴 → (ω ∈ On → 𝐴 ≈ suc 𝐴)))
57	56	imp 445	. 2 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (ω ∈ On → 𝐴 ≈ suc 𝐴))
58	9, 57	mpd 15	1 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ≈ suc 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177   class class class wbr 4653  Ord word 5722  Oncon0 5723  Lim wlim 5724  suc csuc 5725  ωcom 7065   ≈ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957

This theorem is referenced by:  cardlim  8798  cardsucinf  8810