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Theorem onomeneq 8697
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 8693 . . . . . . . . 9 (𝐵 ∈ ω → ¬ 𝐵 ≈ suc 𝐵)
21ad2antlr 726 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ 𝐵 ≈ suc 𝐵)
3 enen1 8645 . . . . . . . . 9 (𝐴𝐵 → (𝐴 ≈ suc 𝐵𝐵 ≈ suc 𝐵))
43adantl 485 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 ≈ suc 𝐵𝐵 ≈ suc 𝐵))
52, 4mtbird 328 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ 𝐴 ≈ suc 𝐵)
6 peano2 7587 . . . . . . . . . . . . . 14 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
7 sssucid 6246 . . . . . . . . . . . . . 14 𝐵 ⊆ suc 𝐵
8 ssdomg 8542 . . . . . . . . . . . . . 14 (suc 𝐵 ∈ ω → (𝐵 ⊆ suc 𝐵𝐵 ≼ suc 𝐵))
96, 7, 8mpisyl 21 . . . . . . . . . . . . 13 (𝐵 ∈ ω → 𝐵 ≼ suc 𝐵)
10 endomtr 8554 . . . . . . . . . . . . 13 ((𝐴𝐵𝐵 ≼ suc 𝐵) → 𝐴 ≼ suc 𝐵)
119, 10sylan2 595 . . . . . . . . . . . 12 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ≼ suc 𝐵)
1211ancoms 462 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴 ≼ suc 𝐵)
1312a1d 25 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (ω ⊆ 𝐴𝐴 ≼ suc 𝐵))
1413adantll 713 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴𝐴 ≼ suc 𝐵))
15 ssel 3935 . . . . . . . . . . . . . . 15 (ω ⊆ 𝐴 → (𝐵 ∈ ω → 𝐵𝐴))
1615com12 32 . . . . . . . . . . . . . 14 (𝐵 ∈ ω → (ω ⊆ 𝐴𝐵𝐴))
1716adantr 484 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴𝐵𝐴))
18 eloni 6179 . . . . . . . . . . . . . 14 (𝐴 ∈ On → Ord 𝐴)
19 ordelsuc 7520 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ Ord 𝐴) → (𝐵𝐴 ↔ suc 𝐵𝐴))
2018, 19sylan2 595 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (𝐵𝐴 ↔ suc 𝐵𝐴))
2117, 20sylibd 242 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
22 ssdomg 8542 . . . . . . . . . . . . 13 (𝐴 ∈ On → (suc 𝐵𝐴 → suc 𝐵𝐴))
2322adantl 485 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (suc 𝐵𝐴 → suc 𝐵𝐴))
2421, 23syld 47 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
2524ancoms 462 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
2625adantr 484 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
2714, 26jcad 516 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴 → (𝐴 ≼ suc 𝐵 ∧ suc 𝐵𝐴)))
28 sbth 8625 . . . . . . . 8 ((𝐴 ≼ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐴 ≈ suc 𝐵)
2927, 28syl6 35 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴𝐴 ≈ suc 𝐵))
305, 29mtod 201 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ ω ⊆ 𝐴)
31 ordom 7574 . . . . . . . . 9 Ord ω
32 ordtri1 6202 . . . . . . . . 9 ((Ord ω ∧ Ord 𝐴) → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
3331, 18, 32sylancr 590 . . . . . . . 8 (𝐴 ∈ On → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
3433con2bid 358 . . . . . . 7 (𝐴 ∈ On → (𝐴 ∈ ω ↔ ¬ ω ⊆ 𝐴))
3534ad2antrr 725 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 ∈ ω ↔ ¬ ω ⊆ 𝐴))
3630, 35mpbird 260 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 ∈ ω)
37 simplr 768 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ ω)
3836, 37jca 515 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 ∈ ω ∧ 𝐵 ∈ ω))
39 nneneq 8688 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
4039biimpa 480 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 = 𝐵)
4138, 40sylancom 591 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 = 𝐵)
4241ex 416 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
43 eqeng 8530 . . 3 (𝐴 ∈ On → (𝐴 = 𝐵𝐴𝐵))
4443adantr 484 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵𝐴𝐵))
4542, 44impbid 215 1 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wss 3908   class class class wbr 5042  Ord word 6168  Oncon0 6169  suc csuc 6171  ωcom 7565  cen 8493  cdom 8494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-om 7566  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499
This theorem is referenced by:  onfin  8698  ficardom  9378  finnisoeu  9528
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