Theorem infensuc 9127

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 7761	. . . . 5 ⊢ ¬ On ∈ V
2		eleq1 2850	. . . . 5 ⊢ (ω = On → (ω ∈ V ↔ On ∈ V))
3	1, 2	mtbiri 329	. . . 4 ⊢ (ω = On → ¬ ω ∈ V)
4		ssexg 5279	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ 𝐴 ∈ On) → ω ∈ V)
5	4	ancoms 462	. . . 4 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ω ∈ V)
6	3, 5	nsyl3 138	. . 3 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ¬ ω = On)
7		omon 7858	. . . 4 ⊢ (ω ∈ On ∨ ω = On)
8	7	ori 872	. . 3 ⊢ (¬ ω ∈ On → ω = On)
9	6, 8	nsyl2 141	. 2 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ω ∈ On)
10		id 22	. . . . . . 7 ⊢ (𝑥 = ω → 𝑥 = ω)
11		suceq 6414	. . . . . . 7 ⊢ (𝑥 = ω → suc 𝑥 = suc ω)
12	10, 11	breq12d 5113	. . . . . 6 ⊢ (𝑥 = ω → (𝑥 ≈ suc 𝑥 ↔ ω ≈ suc ω))
13		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
14		suceq 6414	. . . . . . 7 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
15	13, 14	breq12d 5113	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑥 ↔ 𝑦 ≈ suc 𝑦))
16		id 22	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
17		suceq 6414	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
18	16, 17	breq12d 5113	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ suc 𝑥 ↔ suc 𝑦 ≈ suc suc 𝑦))
19		id 22	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
20		suceq 6414	. . . . . . 7 ⊢ (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
21	19, 20	breq12d 5113	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ suc 𝑥 ↔ 𝐴 ≈ suc 𝐴))
22		limom 7862	. . . . . . 7 ⊢ Lim ω
23	22	limensuci 9125	. . . . . 6 ⊢ (ω ∈ On → ω ≈ suc ω)
24		vex 3458	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
25	24	sucex 7789	. . . . . . . . . 10 ⊢ suc 𝑦 ∈ V
26		en2sn 9022	. . . . . . . . . 10 ⊢ ((𝑦 ∈ V ∧ suc 𝑦 ∈ V) → {𝑦} ≈ {suc 𝑦})
27	24, 25, 26	mp2an 702	. . . . . . . . 9 ⊢ {𝑦} ≈ {suc 𝑦}
28		eloni 6356	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ On → Ord 𝑦)
29		ordirr 6364	. . . . . . . . . . . . 13 ⊢ (Ord 𝑦 → ¬ 𝑦 ∈ 𝑦)
30	28, 29	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → ¬ 𝑦 ∈ 𝑦)
31		disjsn 4670	. . . . . . . . . . . 12 ⊢ ((𝑦 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑦)
32	30, 31	sylibr 236	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (𝑦 ∩ {𝑦}) = ∅)
33		eloni 6356	. . . . . . . . . . . . 13 ⊢ (suc 𝑦 ∈ On → Ord suc 𝑦)
34		ordirr 6364	. . . . . . . . . . . . 13 ⊢ (Ord suc 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
35	33, 34	syl 17	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ∈ On → ¬ suc 𝑦 ∈ suc 𝑦)
36		onsucb 7797	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On ↔ suc 𝑦 ∈ On)
37		disjsn 4670	. . . . . . . . . . . 12 ⊢ ((suc 𝑦 ∩ {suc 𝑦}) = ∅ ↔ ¬ suc 𝑦 ∈ suc 𝑦)
38	35, 36, 37	3imtr4i 294	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (suc 𝑦 ∩ {suc 𝑦}) = ∅)
39	32, 38	jca 519	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅))
40		unen 9026	. . . . . . . . . . . 12 ⊢ (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (suc 𝑦 ∪ {suc 𝑦}))
41		df-suc 6352	. . . . . . . . . . . 12 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
42		df-suc 6352	. . . . . . . . . . . 12 ⊢ suc suc 𝑦 = (suc 𝑦 ∪ {suc 𝑦})
43	40, 41, 42	3brtr4g 5134	. . . . . . . . . . 11 ⊢ (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → suc 𝑦 ≈ suc suc 𝑦)
44	43	ex 416	. . . . . . . . . 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 701	. . . . . . . 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 736	. . . . . 6 ⊢ (((𝑦 ∈ On ∧ ω ∈ On) ∧ ω ⊆ 𝑦) → (𝑦 ≈ suc 𝑦 → suc 𝑦 ≈ suc suc 𝑦))
49		vex 3458	. . . . . . . . 9 ⊢ 𝑥 ∈ V
50		limensuc 9126	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ≈ suc 𝑥)
51	49, 50	mpan 700	. . . . . . . 8 ⊢ (Lim 𝑥 → 𝑥 ≈ suc 𝑥)
52	51	ad2antrr 736	. . . . . . 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 7841	. . . . 5 ⊢ (((𝐴 ∈ On ∧ ω ∈ On) ∧ ω ⊆ 𝐴) → 𝐴 ≈ suc 𝐴)
55	54	exp31 423	. . . 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 410	. 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 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  Vcvv 3454   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {csn 4582   class class class wbr 5100  Ord word 6345  Oncon0 6346  Lim wlim 6347  suc csuc 6348  ωcom 7846   ≈ cen 8924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-om 7847  df-er 8678  df-en 8928  df-dom 8929

This theorem is referenced by:  cardlim  9930  cardsucinf  9942