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Theorem infensuc 8345
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 7182 . . . . 5 ¬ On ∈ V
2 eleq1 2832 . . . . 5 (ω = On → (ω ∈ V ↔ On ∈ V))
31, 2mtbiri 318 . . . 4 (ω = On → ¬ ω ∈ V)
4 ssexg 4965 . . . . 5 ((ω ⊆ 𝐴𝐴 ∈ On) → ω ∈ V)
54ancoms 450 . . . 4 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ω ∈ V)
63, 5nsyl3 135 . . 3 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ¬ ω = On)
7 omon 7274 . . . 4 (ω ∈ On ∨ ω = On)
87ori 887 . . 3 (¬ ω ∈ On → ω = On)
96, 8nsyl2 144 . 2 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → ω ∈ On)
10 id 22 . . . . . . 7 (𝑥 = ω → 𝑥 = ω)
11 suceq 5973 . . . . . . 7 (𝑥 = ω → suc 𝑥 = suc ω)
1210, 11breq12d 4822 . . . . . 6 (𝑥 = ω → (𝑥 ≈ suc 𝑥 ↔ ω ≈ suc ω))
13 id 22 . . . . . . 7 (𝑥 = 𝑦𝑥 = 𝑦)
14 suceq 5973 . . . . . . 7 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
1513, 14breq12d 4822 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑥𝑦 ≈ suc 𝑦))
16 id 22 . . . . . . 7 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
17 suceq 5973 . . . . . . 7 (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
1816, 17breq12d 4822 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥 ≈ suc 𝑥 ↔ suc 𝑦 ≈ suc suc 𝑦))
19 id 22 . . . . . . 7 (𝑥 = 𝐴𝑥 = 𝐴)
20 suceq 5973 . . . . . . 7 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
2119, 20breq12d 4822 . . . . . 6 (𝑥 = 𝐴 → (𝑥 ≈ suc 𝑥𝐴 ≈ suc 𝐴))
22 limom 7278 . . . . . . 7 Lim ω
2322limensuci 8343 . . . . . 6 (ω ∈ On → ω ≈ suc ω)
24 vex 3353 . . . . . . . . . 10 𝑦 ∈ V
2524sucex 7209 . . . . . . . . . 10 suc 𝑦 ∈ V
26 en2sn 8244 . . . . . . . . . 10 ((𝑦 ∈ V ∧ suc 𝑦 ∈ V) → {𝑦} ≈ {suc 𝑦})
2724, 25, 26mp2an 683 . . . . . . . . 9 {𝑦} ≈ {suc 𝑦}
28 eloni 5918 . . . . . . . . . . . . 13 (𝑦 ∈ On → Ord 𝑦)
29 ordirr 5926 . . . . . . . . . . . . 13 (Ord 𝑦 → ¬ 𝑦𝑦)
3028, 29syl 17 . . . . . . . . . . . 12 (𝑦 ∈ On → ¬ 𝑦𝑦)
31 disjsn 4402 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦𝑦)
3230, 31sylibr 225 . . . . . . . . . . 11 (𝑦 ∈ On → (𝑦 ∩ {𝑦}) = ∅)
33 eloni 5918 . . . . . . . . . . . . 13 (suc 𝑦 ∈ On → Ord suc 𝑦)
34 ordirr 5926 . . . . . . . . . . . . 13 (Ord suc 𝑦 → ¬ suc 𝑦 ∈ suc 𝑦)
3533, 34syl 17 . . . . . . . . . . . 12 (suc 𝑦 ∈ On → ¬ suc 𝑦 ∈ suc 𝑦)
36 sucelon 7215 . . . . . . . . . . . 12 (𝑦 ∈ On ↔ suc 𝑦 ∈ On)
37 disjsn 4402 . . . . . . . . . . . 12 ((suc 𝑦 ∩ {suc 𝑦}) = ∅ ↔ ¬ suc 𝑦 ∈ suc 𝑦)
3835, 36, 373imtr4i 283 . . . . . . . . . . 11 (𝑦 ∈ On → (suc 𝑦 ∩ {suc 𝑦}) = ∅)
3932, 38jca 507 . . . . . . . . . 10 (𝑦 ∈ On → ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅))
40 unen 8247 . . . . . . . . . . . 12 (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → (𝑦 ∪ {𝑦}) ≈ (suc 𝑦 ∪ {suc 𝑦}))
41 df-suc 5914 . . . . . . . . . . . 12 suc 𝑦 = (𝑦 ∪ {𝑦})
42 df-suc 5914 . . . . . . . . . . . 12 suc suc 𝑦 = (suc 𝑦 ∪ {suc 𝑦})
4340, 41, 423brtr4g 4843 . . . . . . . . . . 11 (((𝑦 ≈ suc 𝑦 ∧ {𝑦} ≈ {suc 𝑦}) ∧ ((𝑦 ∩ {𝑦}) = ∅ ∧ (suc 𝑦 ∩ {suc 𝑦}) = ∅)) → suc 𝑦 ≈ suc suc 𝑦)
4443ex 401 . . . . . . . . . 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 682 . . . . . . . 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 717 . . . . . 6 (((𝑦 ∈ On ∧ ω ∈ On) ∧ ω ⊆ 𝑦) → (𝑦 ≈ suc 𝑦 → suc 𝑦 ≈ suc suc 𝑦))
49 vex 3353 . . . . . . . . 9 𝑥 ∈ V
50 limensuc 8344 . . . . . . . . 9 ((𝑥 ∈ V ∧ Lim 𝑥) → 𝑥 ≈ suc 𝑥)
5149, 50mpan 681 . . . . . . . 8 (Lim 𝑥𝑥 ≈ suc 𝑥)
5251ad2antrr 717 . . . . . . 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 7258 . . . . 5 (((𝐴 ∈ On ∧ ω ∈ On) ∧ ω ⊆ 𝐴) → 𝐴 ≈ suc 𝐴)
5554exp31 410 . . . 4 (𝐴 ∈ On → (ω ∈ On → (ω ⊆ 𝐴𝐴 ≈ suc 𝐴)))
5655com23 86 . . 3 (𝐴 ∈ On → (ω ⊆ 𝐴 → (ω ∈ On → 𝐴 ≈ suc 𝐴)))
5756imp 395 . 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 384   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  cun 3730  cin 3731  wss 3732  c0 4079  {csn 4334   class class class wbr 4809  Ord word 5907  Oncon0 5908  Lim wlim 5909  suc csuc 5910  ωcom 7263  cen 8157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-1o 7764  df-er 7947  df-en 8161  df-dom 8162
This theorem is referenced by:  cardlim  9049  cardsucinf  9061
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