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Theorem infensuc 8179
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 7026 . . . . 5 ¬ On ∈ V
2 eleq1 2718 . . . . 5 (ω = On → (ω ∈ V ↔ On ∈ V))
31, 2mtbiri 316 . . . 4 (ω = On → ¬ ω ∈ V)
4 ssexg 4837 . . . . 5 ((ω ⊆ 𝐴𝐴 ∈ On) → ω ∈ V)
54ancoms 468 . . . 4 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ω ∈ V)
63, 5nsyl3 133 . . 3 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ¬ ω = On)
7 omon 7118 . . . 4 (ω ∈ On ∨ ω = On)
87ori 389 . . 3 (¬ ω ∈ On → ω = On)
96, 8nsyl2 142 . 2 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ω ∈ On)
10 id 22 . . . . . . 7 (𝑥 = ω → 𝑥 = ω)
11 suceq 5828 . . . . . . 7 (𝑥 = ω → suc 𝑥 = suc ω)
1210, 11breq12d 4698 . . . . . 6 (𝑥 = ω → (𝑥 ≈ suc 𝑥 ↔ ω ≈ suc ω))
13 id 22 . . . . . . 7 (𝑥 = 𝑦𝑥 = 𝑦)
14 suceq 5828 . . . . . . 7 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
1513, 14breq12d 4698 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑥𝑦 ≈ suc 𝑦))
16 id 22 . . . . . . 7 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
17 suceq 5828 . . . . . . 7 (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
1816, 17breq12d 4698 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥 ≈ suc 𝑥 ↔ suc 𝑦 ≈ suc suc 𝑦))
19 id 22 . . . . . . 7 (𝑥 = 𝐴𝑥 = 𝐴)
20 suceq 5828 . . . . . . 7 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
2119, 20breq12d 4698 . . . . . 6 (𝑥 = 𝐴 → (𝑥 ≈ suc 𝑥𝐴 ≈ suc 𝐴))
22 limom 7122 . . . . . . 7 Lim ω
2322limensuci 8177 . . . . . 6 (ω ∈ On → ω ≈ suc ω)
24 vex 3234 . . . . . . . . . 10 𝑦 ∈ V
2524sucex 7053 . . . . . . . . . 10 suc 𝑦 ∈ V
26 en2sn 8078 . . . . . . . . . 10 ((𝑦 ∈ V ∧ suc 𝑦 ∈ V) → {𝑦} ≈ {suc 𝑦})
2724, 25, 26mp2an 708 . . . . . . . . 9 {𝑦} ≈ {suc 𝑦}
28 eloni 5771 . . . . . . . . . . . . 13 (𝑦 ∈ On → Ord 𝑦)
29 ordirr 5779 . . . . . . . . . . . . 13 (Ord 𝑦 → ¬ 𝑦𝑦)
3028, 29syl 17 . . . . . . . . . . . 12 (𝑦 ∈ On → ¬ 𝑦𝑦)
31 disjsn 4278 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝑦)
3230, 31sylibr 224 . . . . . . . . . . 11 (𝑦 ∈ On → (𝑦 ∩ {𝑦}) = ∅)
33 eloni 5771 . . . . . . . . . . . . 13 (suc 𝑦 ∈ On → Ord suc 𝑦)
34 ordirr 5779 . . . . . . . . . . . . 13 (Ord suc 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
3533, 34syl 17 . . . . . . . . . . . 12 (suc 𝑦 ∈ On → ¬ suc 𝑦 ∈ suc 𝑦)
36 sucelon 7059 . . . . . . . . . . . 12 (𝑦 ∈ On ↔ suc 𝑦 ∈ On)
37 disjsn 4278 . . . . . . . . . . . 12 ((suc 𝑦 ∩ {suc 𝑦}) = ∅ ↔ ¬ suc 𝑦 ∈ suc 𝑦)
3835, 36, 373imtr4i 281 . . . . . . . . . . 11 (𝑦 ∈ On → (suc 𝑦 ∩ {suc 𝑦}) = ∅)
3932, 38jca 553 . . . . . . . . . 10 (𝑦 ∈ On → ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅))
40 unen 8081 . . . . . . . . . . . 12 (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (suc 𝑦 ∪ {suc 𝑦}))
41 df-suc 5767 . . . . . . . . . . . 12 suc 𝑦 = (𝑦 ∪ {𝑦})
42 df-suc 5767 . . . . . . . . . . . 12 suc suc 𝑦 = (suc 𝑦 ∪ {suc 𝑦})
4340, 41, 423brtr4g 4719 . . . . . . . . . . 11 (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → suc 𝑦 ≈ suc suc 𝑦)
4443ex 449 . . . . . . . . . 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 707 . . . . . . . 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 762 . . . . . 6 (((𝑦 ∈ On ∧ ω ∈ On) ∧ ω ⊆ 𝑦) → (𝑦 ≈ suc 𝑦 → suc 𝑦 ≈ suc suc 𝑦))
49 vex 3234 . . . . . . . . 9 𝑥 ∈ V
50 limensuc 8178 . . . . . . . . 9 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ≈ suc 𝑥)
5149, 50mpan 706 . . . . . . . 8 (Lim 𝑥𝑥 ≈ suc 𝑥)
5251ad2antrr 762 . . . . . . 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 7102 . . . . 5 (((𝐴 ∈ On ∧ ω ∈ On) ∧ ω ⊆ 𝐴) → 𝐴 ≈ suc 𝐴)
5554exp31 629 . . . 4 (𝐴 ∈ On → (ω ∈ On → (ω ⊆ 𝐴𝐴 ≈ suc 𝐴)))
5655com23 86 . . 3 (𝐴 ∈ On → (ω ⊆ 𝐴 → (ω ∈ On → 𝐴 ≈ suc 𝐴)))
5756imp 444 . 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 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210   class class class wbr 4685  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763  ωcom 7107  cen 7994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999
This theorem is referenced by:  cardlim  8836  cardsucinf  8848
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