Theorem infensuc 9174

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 7777	. . . . 5 ⊢ ¬ On ∈ V
2		eleq1 2823	. . . . 5 ⊢ (ω = On → (ω ∈ V ↔ On ∈ V))
3	1, 2	mtbiri 327	. . . 4 ⊢ (ω = On → ¬ ω ∈ V)
4		ssexg 5298	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ 𝐴 ∈ On) → ω ∈ V)
5	4	ancoms 458	. . . 4 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ω ∈ V)
6	3, 5	nsyl3 138	. . 3 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ¬ ω = On)
7		omon 7878	. . . 4 ⊢ (ω ∈ On ∨ ω = On)
8	7	ori 861	. . 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 6424	. . . . . . 7 ⊢ (𝑥 = ω → suc 𝑥 = suc ω)
12	10, 11	breq12d 5137	. . . . . 6 ⊢ (𝑥 = ω → (𝑥 ≈ suc 𝑥 ↔ ω ≈ suc ω))
13		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
14		suceq 6424	. . . . . . 7 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
15	13, 14	breq12d 5137	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑥 ↔ 𝑦 ≈ suc 𝑦))
16		id 22	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
17		suceq 6424	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
18	16, 17	breq12d 5137	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ suc 𝑥 ↔ suc 𝑦 ≈ suc suc 𝑦))
19		id 22	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
20		suceq 6424	. . . . . . 7 ⊢ (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
21	19, 20	breq12d 5137	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ suc 𝑥 ↔ 𝐴 ≈ suc 𝐴))
22		limom 7882	. . . . . . 7 ⊢ Lim ω
23	22	limensuci 9172	. . . . . 6 ⊢ (ω ∈ On → ω ≈ suc ω)
24		vex 3468	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
25	24	sucex 7805	. . . . . . . . . 10 ⊢ suc 𝑦 ∈ V
26		en2sn 9060	. . . . . . . . . 10 ⊢ ((𝑦 ∈ V ∧ suc 𝑦 ∈ V) → {𝑦} ≈ {suc 𝑦})
27	24, 25, 26	mp2an 692	. . . . . . . . 9 ⊢ {𝑦} ≈ {suc 𝑦}
28		eloni 6367	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ On → Ord 𝑦)
29		ordirr 6375	. . . . . . . . . . . . 13 ⊢ (Ord 𝑦 → ¬ 𝑦 ∈ 𝑦)
30	28, 29	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → ¬ 𝑦 ∈ 𝑦)
31		disjsn 4692	. . . . . . . . . . . 12 ⊢ ((𝑦 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑦)
32	30, 31	sylibr 234	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (𝑦 ∩ {𝑦}) = ∅)
33		eloni 6367	. . . . . . . . . . . . 13 ⊢ (suc 𝑦 ∈ On → Ord suc 𝑦)
34		ordirr 6375	. . . . . . . . . . . . 13 ⊢ (Ord suc 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
35	33, 34	syl 17	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ∈ On → ¬ suc 𝑦 ∈ suc 𝑦)
36		onsucb 7816	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On ↔ suc 𝑦 ∈ On)
37		disjsn 4692	. . . . . . . . . . . 12 ⊢ ((suc 𝑦 ∩ {suc 𝑦}) = ∅ ↔ ¬ suc 𝑦 ∈ suc 𝑦)
38	35, 36, 37	3imtr4i 292	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (suc 𝑦 ∩ {suc 𝑦}) = ∅)
39	32, 38	jca 511	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅))
40		unen 9065	. . . . . . . . . . . 12 ⊢ (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (suc 𝑦 ∪ {suc 𝑦}))
41		df-suc 6363	. . . . . . . . . . . 12 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
42		df-suc 6363	. . . . . . . . . . . 12 ⊢ suc suc 𝑦 = (suc 𝑦 ∪ {suc 𝑦})
43	40, 41, 42	3brtr4g 5158	. . . . . . . . . . 11 ⊢ (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → suc 𝑦 ≈ suc suc 𝑦)
44	43	ex 412	. . . . . . . . . 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 691	. . . . . . . 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 726	. . . . . 6 ⊢ (((𝑦 ∈ On ∧ ω ∈ On) ∧ ω ⊆ 𝑦) → (𝑦 ≈ suc 𝑦 → suc 𝑦 ≈ suc suc 𝑦))
49		vex 3468	. . . . . . . . 9 ⊢ 𝑥 ∈ V
50		limensuc 9173	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ≈ suc 𝑥)
51	49, 50	mpan 690	. . . . . . . 8 ⊢ (Lim 𝑥 → 𝑥 ≈ suc 𝑥)
52	51	ad2antrr 726	. . . . . . 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 7861	. . . . 5 ⊢ (((𝐴 ∈ On ∧ ω ∈ On) ∧ ω ⊆ 𝐴) → 𝐴 ≈ suc 𝐴)
55	54	exp31 419	. . . 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 406	. 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 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464   ∪ cun 3929   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  {csn 4606   class class class wbr 5124  Ord word 6356  Oncon0 6357  Lim wlim 6358  suc csuc 6359  ωcom 7866   ≈ cen 8961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7867  df-er 8724  df-en 8965  df-dom 8966

This theorem is referenced by:  cardlim  9991  cardsucinf  10003