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Theorem infensuc 9180
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.
StepHypRef Expression
1 onprc 7781 . . . . 5 ¬ On ∈ V
2 eleq1 2813 . . . . 5 (ω = On → (ω ∈ V ↔ On ∈ V))
31, 2mtbiri 326 . . . 4 (ω = On → ¬ ω ∈ V)
4 ssexg 5324 . . . . 5 ((ω ⊆ 𝐴𝐴 ∈ On) → ω ∈ V)
54ancoms 457 . . . 4 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ω ∈ V)
63, 5nsyl3 138 . . 3 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ¬ ω = On)
7 omon 7883 . . . 4 (ω ∈ On ∨ ω = On)
87ori 859 . . 3 (¬ ω ∈ On → ω = On)
96, 8nsyl2 141 . 2 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ω ∈ On)
10 id 22 . . . . . . 7 (𝑥 = ω → 𝑥 = ω)
11 suceq 6437 . . . . . . 7 (𝑥 = ω → suc 𝑥 = suc ω)
1210, 11breq12d 5162 . . . . . 6 (𝑥 = ω → (𝑥 ≈ suc 𝑥 ↔ ω ≈ suc ω))
13 id 22 . . . . . . 7 (𝑥 = 𝑦𝑥 = 𝑦)
14 suceq 6437 . . . . . . 7 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
1513, 14breq12d 5162 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑥𝑦 ≈ suc 𝑦))
16 id 22 . . . . . . 7 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
17 suceq 6437 . . . . . . 7 (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
1816, 17breq12d 5162 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥 ≈ suc 𝑥 ↔ suc 𝑦 ≈ suc suc 𝑦))
19 id 22 . . . . . . 7 (𝑥 = 𝐴𝑥 = 𝐴)
20 suceq 6437 . . . . . . 7 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
2119, 20breq12d 5162 . . . . . 6 (𝑥 = 𝐴 → (𝑥 ≈ suc 𝑥𝐴 ≈ suc 𝐴))
22 limom 7887 . . . . . . 7 Lim ω
2322limensuci 9178 . . . . . 6 (ω ∈ On → ω ≈ suc ω)
24 vex 3465 . . . . . . . . . 10 𝑦 ∈ V
2524sucex 7810 . . . . . . . . . 10 suc 𝑦 ∈ V
26 en2sn 9066 . . . . . . . . . 10 ((𝑦 ∈ V ∧ suc 𝑦 ∈ V) → {𝑦} ≈ {suc 𝑦})
2724, 25, 26mp2an 690 . . . . . . . . 9 {𝑦} ≈ {suc 𝑦}
28 eloni 6381 . . . . . . . . . . . . 13 (𝑦 ∈ On → Ord 𝑦)
29 ordirr 6389 . . . . . . . . . . . . 13 (Ord 𝑦 → ¬ 𝑦𝑦)
3028, 29syl 17 . . . . . . . . . . . 12 (𝑦 ∈ On → ¬ 𝑦𝑦)
31 disjsn 4717 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝑦)
3230, 31sylibr 233 . . . . . . . . . . 11 (𝑦 ∈ On → (𝑦 ∩ {𝑦}) = ∅)
33 eloni 6381 . . . . . . . . . . . . 13 (suc 𝑦 ∈ On → Ord suc 𝑦)
34 ordirr 6389 . . . . . . . . . . . . 13 (Ord suc 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
3533, 34syl 17 . . . . . . . . . . . 12 (suc 𝑦 ∈ On → ¬ suc 𝑦 ∈ suc 𝑦)
36 onsucb 7821 . . . . . . . . . . . 12 (𝑦 ∈ On ↔ suc 𝑦 ∈ On)
37 disjsn 4717 . . . . . . . . . . . 12 ((suc 𝑦 ∩ {suc 𝑦}) = ∅ ↔ ¬ suc 𝑦 ∈ suc 𝑦)
3835, 36, 373imtr4i 291 . . . . . . . . . . 11 (𝑦 ∈ On → (suc 𝑦 ∩ {suc 𝑦}) = ∅)
3932, 38jca 510 . . . . . . . . . 10 (𝑦 ∈ On → ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅))
40 unen 9071 . . . . . . . . . . . 12 (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (suc 𝑦 ∪ {suc 𝑦}))
41 df-suc 6377 . . . . . . . . . . . 12 suc 𝑦 = (𝑦 ∪ {𝑦})
42 df-suc 6377 . . . . . . . . . . . 12 suc suc 𝑦 = (suc 𝑦 ∪ {suc 𝑦})
4340, 41, 423brtr4g 5183 . . . . . . . . . . 11 (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → suc 𝑦 ≈ suc suc 𝑦)
4443ex 411 . . . . . . . . . 10 ((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) → (((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅) → suc 𝑦 ≈ suc suc 𝑦))
4539, 44syl5 34 . . . . . . . . 9 ((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) → (𝑦 ∈ On → suc 𝑦 ≈ suc suc 𝑦))
4627, 45mpan2 689 . . . . . . . 8 (𝑦 ≈ suc 𝑦 → (𝑦 ∈ On → suc 𝑦 ≈ suc suc 𝑦))
4746com12 32 . . . . . . 7 (𝑦 ∈ On → (𝑦 ≈ suc 𝑦 → suc 𝑦 ≈ suc suc 𝑦))
4847ad2antrr 724 . . . . . 6 (((𝑦 ∈ On ∧ ω ∈ On) ∧ ω ⊆ 𝑦) → (𝑦 ≈ suc 𝑦 → suc 𝑦 ≈ suc suc 𝑦))
49 vex 3465 . . . . . . . . 9 𝑥 ∈ V
50 limensuc 9179 . . . . . . . . 9 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ≈ suc 𝑥)
5149, 50mpan 688 . . . . . . . 8 (Lim 𝑥𝑥 ≈ suc 𝑥)
5251ad2antrr 724 . . . . . . 7 (((Lim 𝑥 ∧ ω ∈ On) ∧ ω ⊆ 𝑥) → 𝑥 ≈ suc 𝑥)
5352a1d 25 . . . . . 6 (((Lim 𝑥 ∧ ω ∈ On) ∧ ω ⊆ 𝑥) → (∀𝑦𝑥 (ω ⊆ 𝑦𝑦 ≈ suc 𝑦) → 𝑥 ≈ suc 𝑥))
5412, 15, 18, 21, 23, 48, 53tfindsg 7866 . . . . 5 (((𝐴 ∈ On ∧ ω ∈ On) ∧ ω ⊆ 𝐴) → 𝐴 ≈ suc 𝐴)
5554exp31 418 . . . 4 (𝐴 ∈ On → (ω ∈ On → (ω ⊆ 𝐴𝐴 ≈ suc 𝐴)))
5655com23 86 . . 3 (𝐴 ∈ On → (ω ⊆ 𝐴 → (ω ∈ On → 𝐴 ≈ suc 𝐴)))
5756imp 405 . 2 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (ω ∈ On → 𝐴 ≈ suc 𝐴))
589, 57mpd 15 1 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ≈ suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3050  Vcvv 3461  cun 3942  cin 3943  wss 3944  c0 4322  {csn 4630   class class class wbr 5149  Ord word 6370  Oncon0 6371  Lim wlim 6372  suc csuc 6373  ωcom 7871  cen 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-om 7872  df-er 8725  df-en 8965  df-dom 8966
This theorem is referenced by:  cardlim  9997  cardsucinf  10009
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