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Theorem onomeneq 8035
 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 8033 . . . . . . . . 9 (𝐵 ∈ ω → ¬ 𝐵 ≈ suc 𝐵)
21ad2antlr 759 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ 𝐵 ≈ suc 𝐵)
3 enen1 7985 . . . . . . . . 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 6978 . . . . . . . . . . . . . 14 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
7 sssucid 5719 . . . . . . . . . . . . . 14 𝐵 ⊆ suc 𝐵
8 ssdomg 7887 . . . . . . . . . . . . . 14 (suc 𝐵 ∈ ω → (𝐵 ⊆ suc 𝐵𝐵 ≼ suc 𝐵))
96, 7, 8mpisyl 21 . . . . . . . . . . . . 13 (𝐵 ∈ ω → 𝐵 ≼ suc 𝐵)
10 endomtr 7900 . . . . . . . . . . . . 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 746 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴𝐴 ≼ suc 𝐵))
15 ssel 3562 . . . . . . . . . . . . . . 15 (ω ⊆ 𝐴 → (𝐵 ∈ ω → 𝐵𝐴))
1615com12 32 . . . . . . . . . . . . . 14 (𝐵 ∈ ω → (ω ⊆ 𝐴𝐵𝐴))
1716adantr 480 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴𝐵𝐴))
18 eloni 5650 . . . . . . . . . . . . . 14 (𝐴 ∈ On → Ord 𝐴)
19 ordelsuc 6912 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ Ord 𝐴) → (𝐵𝐴 ↔ suc 𝐵𝐴))
2018, 19sylan2 490 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (𝐵𝐴 ↔ suc 𝐵𝐴))
2117, 20sylibd 228 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
22 ssdomg 7887 . . . . . . . . . . . . 13 (𝐴 ∈ On → (suc 𝐵𝐴 → suc 𝐵𝐴))
2322adantl 481 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (suc 𝐵𝐴 → suc 𝐵𝐴))
2421, 23syld 46 . . . . . . . . . . 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 7965 . . . . . . . 8 ((𝐴 ≼ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐴 ≈ suc 𝐵)
2927, 28syl6 34 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴𝐴 ≈ suc 𝐵))
305, 29mtod 188 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ ω ⊆ 𝐴)
31 ordom 6966 . . . . . . . . 9 Ord ω
32 ordtri1 5673 . . . . . . . . 9 ((Ord ω ∧ Ord 𝐴) → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
3331, 18, 32sylancr 694 . . . . . . . 8 (𝐴 ∈ On → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
3433con2bid 343 . . . . . . 7 (𝐴 ∈ On → (𝐴 ∈ ω ↔ ¬ ω ⊆ 𝐴))
3534ad2antrr 758 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 ∈ ω ↔ ¬ ω ⊆ 𝐴))
3630, 35mpbird 246 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 ∈ ω)
37 simplr 788 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ ω)
3836, 37jca 553 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 ∈ ω ∧ 𝐵 ∈ ω))
39 nneneq 8028 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
4039biimpa 500 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 = 𝐵)
4138, 40sylancom 698 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 = 𝐵)
4241ex 449 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
43 eqeng 7875 . . 3 (𝐴 ∈ On → (𝐴 = 𝐵𝐴𝐵))
4443adantr 480 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵𝐴𝐵))
4542, 44impbid 201 1 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540   class class class wbr 4583  Ord word 5639  Oncon0 5640  suc csuc 5642  ωcom 6957   ≈ cen 7838   ≼ cdom 7839 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844 This theorem is referenced by:  onfin  8036  ficardom  8670  finnisoeu  8819
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