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Theorem infensuc 8671
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 7475 . . . . 5 ¬ On ∈ V
2 eleq1 2898 . . . . 5 (ω = On → (ω ∈ V ↔ On ∈ V))
31, 2mtbiri 329 . . . 4 (ω = On → ¬ ω ∈ V)
4 ssexg 5201 . . . . 5 ((ω ⊆ 𝐴𝐴 ∈ On) → ω ∈ V)
54ancoms 461 . . . 4 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ω ∈ V)
63, 5nsyl3 140 . . 3 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ¬ ω = On)
7 omon 7567 . . . 4 (ω ∈ On ∨ ω = On)
87ori 857 . . 3 (¬ ω ∈ On → ω = On)
96, 8nsyl2 143 . 2 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ω ∈ On)
10 id 22 . . . . . . 7 (𝑥 = ω → 𝑥 = ω)
11 suceq 6230 . . . . . . 7 (𝑥 = ω → suc 𝑥 = suc ω)
1210, 11breq12d 5053 . . . . . 6 (𝑥 = ω → (𝑥 ≈ suc 𝑥 ↔ ω ≈ suc ω))
13 id 22 . . . . . . 7 (𝑥 = 𝑦𝑥 = 𝑦)
14 suceq 6230 . . . . . . 7 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
1513, 14breq12d 5053 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑥𝑦 ≈ suc 𝑦))
16 id 22 . . . . . . 7 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
17 suceq 6230 . . . . . . 7 (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
1816, 17breq12d 5053 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥 ≈ suc 𝑥 ↔ suc 𝑦 ≈ suc suc 𝑦))
19 id 22 . . . . . . 7 (𝑥 = 𝐴𝑥 = 𝐴)
20 suceq 6230 . . . . . . 7 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
2119, 20breq12d 5053 . . . . . 6 (𝑥 = 𝐴 → (𝑥 ≈ suc 𝑥𝐴 ≈ suc 𝐴))
22 limom 7571 . . . . . . 7 Lim ω
2322limensuci 8669 . . . . . 6 (ω ∈ On → ω ≈ suc ω)
24 vex 3476 . . . . . . . . . 10 𝑦 ∈ V
2524sucex 7502 . . . . . . . . . 10 suc 𝑦 ∈ V
26 en2sn 8569 . . . . . . . . . 10 ((𝑦 ∈ V ∧ suc 𝑦 ∈ V) → {𝑦} ≈ {suc 𝑦})
2724, 25, 26mp2an 690 . . . . . . . . 9 {𝑦} ≈ {suc 𝑦}
28 eloni 6175 . . . . . . . . . . . . 13 (𝑦 ∈ On → Ord 𝑦)
29 ordirr 6183 . . . . . . . . . . . . 13 (Ord 𝑦 → ¬ 𝑦𝑦)
3028, 29syl 17 . . . . . . . . . . . 12 (𝑦 ∈ On → ¬ 𝑦𝑦)
31 disjsn 4621 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝑦)
3230, 31sylibr 236 . . . . . . . . . . 11 (𝑦 ∈ On → (𝑦 ∩ {𝑦}) = ∅)
33 eloni 6175 . . . . . . . . . . . . 13 (suc 𝑦 ∈ On → Ord suc 𝑦)
34 ordirr 6183 . . . . . . . . . . . . 13 (Ord suc 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
3533, 34syl 17 . . . . . . . . . . . 12 (suc 𝑦 ∈ On → ¬ suc 𝑦 ∈ suc 𝑦)
36 sucelon 7508 . . . . . . . . . . . 12 (𝑦 ∈ On ↔ suc 𝑦 ∈ On)
37 disjsn 4621 . . . . . . . . . . . 12 ((suc 𝑦 ∩ {suc 𝑦}) = ∅ ↔ ¬ suc 𝑦 ∈ suc 𝑦)
3835, 36, 373imtr4i 294 . . . . . . . . . . 11 (𝑦 ∈ On → (suc 𝑦 ∩ {suc 𝑦}) = ∅)
3932, 38jca 514 . . . . . . . . . 10 (𝑦 ∈ On → ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅))
40 unen 8572 . . . . . . . . . . . 12 (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (suc 𝑦 ∪ {suc 𝑦}))
41 df-suc 6171 . . . . . . . . . . . 12 suc 𝑦 = (𝑦 ∪ {𝑦})
42 df-suc 6171 . . . . . . . . . . . 12 suc suc 𝑦 = (suc 𝑦 ∪ {suc 𝑦})
4340, 41, 423brtr4g 5074 . . . . . . . . . . 11 (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → suc 𝑦 ≈ suc suc 𝑦)
4443ex 415 . . . . . . . . . 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 3476 . . . . . . . . 9 𝑥 ∈ V
50 limensuc 8670 . . . . . . . . 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 7551 . . . . 5 (((𝐴 ∈ On ∧ ω ∈ On) ∧ ω ⊆ 𝐴) → 𝐴 ≈ suc 𝐴)
5554exp31 422 . . . 4 (𝐴 ∈ On → (ω ∈ On → (ω ⊆ 𝐴𝐴 ≈ suc 𝐴)))
5655com23 86 . . 3 (𝐴 ∈ On → (ω ⊆ 𝐴 → (ω ∈ On → 𝐴 ≈ suc 𝐴)))
5756imp 409 . 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 398   = wceq 1537  wcel 2114  wral 3125  Vcvv 3473  cun 3910  cin 3911  wss 3912  c0 4267  {csn 4541   class class class wbr 5040  Ord word 6164  Oncon0 6165  Lim wlim 6166  suc csuc 6167  ωcom 7556  cen 8482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-pss 3930  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4813  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5434  df-eprel 5439  df-po 5448  df-so 5449  df-fr 5488  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-ord 6168  df-on 6169  df-lim 6170  df-suc 6171  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-om 7557  df-1o 8078  df-er 8265  df-en 8486  df-dom 8487
This theorem is referenced by:  cardlim  9377  cardsucinf  9389
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