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Theorem onomeneq 8135
Description: An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse. (Contributed by NM, 26-Jul-2004.)
Assertion
Ref Expression
onomeneq ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem onomeneq
StepHypRef Expression
1 php5 8133 . . . . . . . . 9 (𝐵 ∈ ω → ¬ 𝐵 ≈ suc 𝐵)
21ad2antlr 762 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ 𝐵 ≈ suc 𝐵)
3 enen1 8085 . . . . . . . . 9 (𝐴𝐵 → (𝐴 ≈ suc 𝐵𝐵 ≈ suc 𝐵))
43adantl 482 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 ≈ suc 𝐵𝐵 ≈ suc 𝐵))
52, 4mtbird 315 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ 𝐴 ≈ suc 𝐵)
6 peano2 7071 . . . . . . . . . . . . . 14 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
7 sssucid 5790 . . . . . . . . . . . . . 14 𝐵 ⊆ suc 𝐵
8 ssdomg 7986 . . . . . . . . . . . . . 14 (suc 𝐵 ∈ ω → (𝐵 ⊆ suc 𝐵𝐵 ≼ suc 𝐵))
96, 7, 8mpisyl 21 . . . . . . . . . . . . 13 (𝐵 ∈ ω → 𝐵 ≼ suc 𝐵)
10 endomtr 7999 . . . . . . . . . . . . 13 ((𝐴𝐵𝐵 ≼ suc 𝐵) → 𝐴 ≼ suc 𝐵)
119, 10sylan2 491 . . . . . . . . . . . 12 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ≼ suc 𝐵)
1211ancoms 469 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴 ≼ suc 𝐵)
1312a1d 25 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (ω ⊆ 𝐴𝐴 ≼ suc 𝐵))
1413adantll 749 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴𝐴 ≼ suc 𝐵))
15 ssel 3589 . . . . . . . . . . . . . . 15 (ω ⊆ 𝐴 → (𝐵 ∈ ω → 𝐵𝐴))
1615com12 32 . . . . . . . . . . . . . 14 (𝐵 ∈ ω → (ω ⊆ 𝐴𝐵𝐴))
1716adantr 481 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴𝐵𝐴))
18 eloni 5721 . . . . . . . . . . . . . 14 (𝐴 ∈ On → Ord 𝐴)
19 ordelsuc 7005 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ Ord 𝐴) → (𝐵𝐴 ↔ suc 𝐵𝐴))
2018, 19sylan2 491 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (𝐵𝐴 ↔ suc 𝐵𝐴))
2117, 20sylibd 229 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
22 ssdomg 7986 . . . . . . . . . . . . 13 (𝐴 ∈ On → (suc 𝐵𝐴 → suc 𝐵𝐴))
2322adantl 482 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (suc 𝐵𝐴 → suc 𝐵𝐴))
2421, 23syld 47 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
2524ancoms 469 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
2625adantr 481 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
2714, 26jcad 555 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴 → (𝐴 ≼ suc 𝐵 ∧ suc 𝐵𝐴)))
28 sbth 8065 . . . . . . . 8 ((𝐴 ≼ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐴 ≈ suc 𝐵)
2927, 28syl6 35 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴𝐴 ≈ suc 𝐵))
305, 29mtod 189 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ ω ⊆ 𝐴)
31 ordom 7059 . . . . . . . . 9 Ord ω
32 ordtri1 5744 . . . . . . . . 9 ((Ord ω ∧ Ord 𝐴) → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
3331, 18, 32sylancr 694 . . . . . . . 8 (𝐴 ∈ On → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
3433con2bid 344 . . . . . . 7 (𝐴 ∈ On → (𝐴 ∈ ω ↔ ¬ ω ⊆ 𝐴))
3534ad2antrr 761 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 ∈ ω ↔ ¬ ω ⊆ 𝐴))
3630, 35mpbird 247 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 ∈ ω)
37 simplr 791 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ ω)
3836, 37jca 554 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 ∈ ω ∧ 𝐵 ∈ ω))
39 nneneq 8128 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
4039biimpa 501 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 = 𝐵)
4138, 40sylancom 700 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 = 𝐵)
4241ex 450 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
43 eqeng 7974 . . 3 (𝐴 ∈ On → (𝐴 = 𝐵𝐴𝐵))
4443adantr 481 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵𝐴𝐵))
4542, 44impbid 202 1 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wss 3567   class class class wbr 4644  Ord word 5710  Oncon0 5711  suc csuc 5713  ωcom 7050  cen 7937  cdom 7938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-om 7051  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943
This theorem is referenced by:  onfin  8136  ficardom  8772  finnisoeu  8921
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