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Theorem onomeneq 8191
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 8189 . . . . . . . . 9 (𝐵 ∈ ω → ¬ 𝐵 ≈ suc 𝐵)
21ad2antlr 763 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ 𝐵 ≈ suc 𝐵)
3 enen1 8141 . . . . . . . . 9 (𝐴𝐵 → (𝐴 ≈ suc 𝐵𝐵 ≈ suc 𝐵))
43adantl 481 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 ≈ suc 𝐵𝐵 ≈ suc 𝐵))
52, 4mtbird 314 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ 𝐴 ≈ suc 𝐵)
6 peano2 7128 . . . . . . . . . . . . . 14 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
7 sssucid 5840 . . . . . . . . . . . . . 14 𝐵 ⊆ suc 𝐵
8 ssdomg 8043 . . . . . . . . . . . . . 14 (suc 𝐵 ∈ ω → (𝐵 ⊆ suc 𝐵𝐵 ≼ suc 𝐵))
96, 7, 8mpisyl 21 . . . . . . . . . . . . 13 (𝐵 ∈ ω → 𝐵 ≼ suc 𝐵)
10 endomtr 8055 . . . . . . . . . . . . 13 ((𝐴𝐵𝐵 ≼ suc 𝐵) → 𝐴 ≼ suc 𝐵)
119, 10sylan2 490 . . . . . . . . . . . 12 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ≼ suc 𝐵)
1211ancoms 468 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴 ≼ suc 𝐵)
1312a1d 25 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (ω ⊆ 𝐴𝐴 ≼ suc 𝐵))
1413adantll 750 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴𝐴 ≼ suc 𝐵))
15 ssel 3630 . . . . . . . . . . . . . . 15 (ω ⊆ 𝐴 → (𝐵 ∈ ω → 𝐵𝐴))
1615com12 32 . . . . . . . . . . . . . 14 (𝐵 ∈ ω → (ω ⊆ 𝐴𝐵𝐴))
1716adantr 480 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴𝐵𝐴))
18 eloni 5771 . . . . . . . . . . . . . 14 (𝐴 ∈ On → Ord 𝐴)
19 ordelsuc 7062 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ Ord 𝐴) → (𝐵𝐴 ↔ suc 𝐵𝐴))
2018, 19sylan2 490 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (𝐵𝐴 ↔ suc 𝐵𝐴))
2117, 20sylibd 229 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
22 ssdomg 8043 . . . . . . . . . . . . 13 (𝐴 ∈ On → (suc 𝐵𝐴 → suc 𝐵𝐴))
2322adantl 481 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (suc 𝐵𝐴 → suc 𝐵𝐴))
2421, 23syld 47 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
2524ancoms 468 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
2625adantr 480 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
2714, 26jcad 554 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴 → (𝐴 ≼ suc 𝐵 ∧ suc 𝐵𝐴)))
28 sbth 8121 . . . . . . . 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 7116 . . . . . . . . 9 Ord ω
32 ordtri1 5794 . . . . . . . . 9 ((Ord ω ∧ Ord 𝐴) → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
3331, 18, 32sylancr 696 . . . . . . . 8 (𝐴 ∈ On → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
3433con2bid 343 . . . . . . 7 (𝐴 ∈ On → (𝐴 ∈ ω ↔ ¬ ω ⊆ 𝐴))
3534ad2antrr 762 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 ∈ ω ↔ ¬ ω ⊆ 𝐴))
3630, 35mpbird 247 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 ∈ ω)
37 simplr 807 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ ω)
3836, 37jca 553 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 ∈ ω ∧ 𝐵 ∈ ω))
39 nneneq 8184 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
4039biimpa 500 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 = 𝐵)
4138, 40sylancom 702 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 = 𝐵)
4241ex 449 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
43 eqeng 8031 . . 3 (𝐴 ∈ On → (𝐴 = 𝐵𝐴𝐵))
4443adantr 480 . 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 383   = wceq 1523  wcel 2030  wss 3607   class class class wbr 4685  Ord word 5760  Oncon0 5761  suc csuc 5763  ωcom 7107  cen 7994  cdom 7995
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-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-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-er 7787  df-en 7998  df-dom 7999  df-sdom 8000
This theorem is referenced by:  onfin  8192  ficardom  8825  finnisoeu  8974
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