Theorem infensuc 9150

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 7783	. . . . 5 ⊢ ¬ On ∈ V
2		eleq1 2853	. . . . 5 ⊢ (ω = On → (ω ∈ V ↔ On ∈ V))
3	1, 2	mtbiri 330	. . . 4 ⊢ (ω = On → ¬ ω ∈ V)
4		ssexg 5292	. . . . 5 ⊢ ((ω ⊆ 𝐴 ∧ 𝐴 ∈ On) → ω ∈ V)
5	4	ancoms 464	. . . 4 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ω ∈ V)
6	3, 5	nsyl3 139	. . 3 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ¬ ω = On)
7		omon 7880	. . . 4 ⊢ (ω ∈ On ∨ ω = On)
8	7	ori 875	. . 3 ⊢ (¬ ω ∈ On → ω = On)
9	6, 8	nsyl2 142	. 2 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ω ∈ On)
10		id 23	. . . . . . 7 ⊢ (𝑥 = ω → 𝑥 = ω)
11		suceq 6433	. . . . . . 7 ⊢ (𝑥 = ω → suc 𝑥 = suc ω)
12	10, 11	breq12d 5124	. . . . . 6 ⊢ (𝑥 = ω → (𝑥 ≈ suc 𝑥 ↔ ω ≈ suc ω))
13		id 23	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
14		suceq 6433	. . . . . . 7 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
15	13, 14	breq12d 5124	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑥 ↔ 𝑦 ≈ suc 𝑦))
16		id 23	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
17		suceq 6433	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
18	16, 17	breq12d 5124	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ suc 𝑥 ↔ suc 𝑦 ≈ suc suc 𝑦))
19		id 23	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
20		suceq 6433	. . . . . . 7 ⊢ (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
21	19, 20	breq12d 5124	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ suc 𝑥 ↔ 𝐴 ≈ suc 𝐴))
22		limom 7884	. . . . . . 7 ⊢ Lim ω
23	22	limensuci 9148	. . . . . 6 ⊢ (ω ∈ On → ω ≈ suc ω)
24		vex 3461	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
25	24	sucex 7811	. . . . . . . . . 10 ⊢ suc 𝑦 ∈ V
26		en2sn 9045	. . . . . . . . . 10 ⊢ ((𝑦 ∈ V ∧ suc 𝑦 ∈ V) → {𝑦} ≈ {suc 𝑦})
27	24, 25, 26	mp2an 705	. . . . . . . . 9 ⊢ {𝑦} ≈ {suc 𝑦}
28		eloni 6374	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ On → Ord 𝑦)
29		ordirr 6382	. . . . . . . . . . . . 13 ⊢ (Ord 𝑦 → ¬ 𝑦 ∈ 𝑦)
30	28, 29	syl 18	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On → ¬ 𝑦 ∈ 𝑦)
31		disjsn 4679	. . . . . . . . . . . 12 ⊢ ((𝑦 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑦)
32	30, 31	sylibr 237	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (𝑦 ∩ {𝑦}) = ∅)
33		eloni 6374	. . . . . . . . . . . . 13 ⊢ (suc 𝑦 ∈ On → Ord suc 𝑦)
34		ordirr 6382	. . . . . . . . . . . . 13 ⊢ (Ord suc 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
35	33, 34	syl 18	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ∈ On → ¬ suc 𝑦 ∈ suc 𝑦)
36		onsucb 7819	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ On ↔ suc 𝑦 ∈ On)
37		disjsn 4679	. . . . . . . . . . . 12 ⊢ ((suc 𝑦 ∩ {suc 𝑦}) = ∅ ↔ ¬ suc 𝑦 ∈ suc 𝑦)
38	35, 36, 37	3imtr4i 295	. . . . . . . . . . 11 ⊢ (𝑦 ∈ On → (suc 𝑦 ∩ {suc 𝑦}) = ∅)
39	32, 38	jca 521	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅))
40		unen 9049	. . . . . . . . . . . 12 ⊢ (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (suc 𝑦 ∪ {suc 𝑦}))
41		df-suc 6370	. . . . . . . . . . . 12 ⊢ suc 𝑦 = (𝑦 ∪ {𝑦})
42		df-suc 6370	. . . . . . . . . . . 12 ⊢ suc suc 𝑦 = (suc 𝑦 ∪ {suc 𝑦})
43	40, 41, 42	3brtr4g 5147	. . . . . . . . . . 11 ⊢ (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → suc 𝑦 ≈ suc suc 𝑦)
44	43	ex 418	. . . . . . . . . 10 ⊢ ((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) → (((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅) → suc 𝑦 ≈ suc suc 𝑦))
45	39, 44	syl5 35	. . . . . . . . 9 ⊢ ((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) → (𝑦 ∈ On → suc 𝑦 ≈ suc suc 𝑦))
46	27, 45	mpan2 704	. . . . . . . 8 ⊢ (𝑦 ≈ suc 𝑦 → (𝑦 ∈ On → suc 𝑦 ≈ suc suc 𝑦))
47	46	com12 33	. . . . . . 7 ⊢ (𝑦 ∈ On → (𝑦 ≈ suc 𝑦 → suc 𝑦 ≈ suc suc 𝑦))
48	47	ad2antrr 739	. . . . . 6 ⊢ (((𝑦 ∈ On ∧ ω ∈ On) ∧ ω ⊆ 𝑦) → (𝑦 ≈ suc 𝑦 → suc 𝑦 ≈ suc suc 𝑦))
49		vex 3461	. . . . . . . . 9 ⊢ 𝑥 ∈ V
50		limensuc 9149	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ≈ suc 𝑥)
51	49, 50	mpan 703	. . . . . . . 8 ⊢ (Lim 𝑥 → 𝑥 ≈ suc 𝑥)
52	51	ad2antrr 739	. . . . . . 7 ⊢ (((Lim 𝑥 ∧ ω ∈ On) ∧ ω ⊆ 𝑥) → 𝑥 ≈ suc 𝑥)
53	52	a1d 26	. . . . . 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 7863	. . . . 5 ⊢ (((𝐴 ∈ On ∧ ω ∈ On) ∧ ω ⊆ 𝐴) → 𝐴 ≈ suc 𝐴)
55	54	exp31 425	. . . 4 ⊢ (𝐴 ∈ On → (ω ∈ On → (ω ⊆ 𝐴 → 𝐴 ≈ suc 𝐴)))
56	55	com23 87	. . 3 ⊢ (𝐴 ∈ On → (ω ⊆ 𝐴 → (ω ∈ On → 𝐴 ≈ suc 𝐴)))
57	56	imp 412	. 2 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (ω ∈ On → 𝐴 ≈ suc 𝐴))
58	9, 57	mpd 16	1 ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ≈ suc 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  Vcvv 3457   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  {csn 4591   class class class wbr 5111  Ord word 6363  Oncon0 6364  Lim wlim 6365  suc csuc 6366  ωcom 7868   ≈ cen 8946

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7869  df-er 8700  df-en 8950  df-dom 8951

This theorem is used by:  cardlim  9974  cardsucinf  9986