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Theorem onomeneqOLD 9266
Description: Obsolete version of onomeneq 9265 as of 29-Nov-2024. (Contributed by NM, 26-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onomeneqOLD ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem onomeneqOLD
StepHypRef Expression
1 php5 9251 . . . . . . . . 9 (𝐵 ∈ ω → ¬ 𝐵 ≈ suc 𝐵)
21ad2antlr 727 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ 𝐵 ≈ suc 𝐵)
3 enen1 9157 . . . . . . . . 9 (𝐴𝐵 → (𝐴 ≈ suc 𝐵𝐵 ≈ suc 𝐵))
43adantl 481 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 ≈ suc 𝐵𝐵 ≈ suc 𝐵))
52, 4mtbird 325 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ 𝐴 ≈ suc 𝐵)
6 peano2 7912 . . . . . . . . . . . . . 14 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
7 sssucid 6464 . . . . . . . . . . . . . 14 𝐵 ⊆ suc 𝐵
8 ssdomg 9040 . . . . . . . . . . . . . 14 (suc 𝐵 ∈ ω → (𝐵 ⊆ suc 𝐵𝐵 ≼ suc 𝐵))
96, 7, 8mpisyl 21 . . . . . . . . . . . . 13 (𝐵 ∈ ω → 𝐵 ≼ suc 𝐵)
10 endomtr 9052 . . . . . . . . . . . . 13 ((𝐴𝐵𝐵 ≼ suc 𝐵) → 𝐴 ≼ suc 𝐵)
119, 10sylan2 593 . . . . . . . . . . . 12 ((𝐴𝐵𝐵 ∈ ω) → 𝐴 ≼ suc 𝐵)
1211ancoms 458 . . . . . . . . . . 11 ((𝐵 ∈ ω ∧ 𝐴𝐵) → 𝐴 ≼ suc 𝐵)
1312a1d 25 . . . . . . . . . 10 ((𝐵 ∈ ω ∧ 𝐴𝐵) → (ω ⊆ 𝐴𝐴 ≼ suc 𝐵))
1413adantll 714 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴𝐴 ≼ suc 𝐵))
15 ssel 3977 . . . . . . . . . . . . . . 15 (ω ⊆ 𝐴 → (𝐵 ∈ ω → 𝐵𝐴))
1615com12 32 . . . . . . . . . . . . . 14 (𝐵 ∈ ω → (ω ⊆ 𝐴𝐵𝐴))
1716adantr 480 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴𝐵𝐴))
18 eloni 6394 . . . . . . . . . . . . . 14 (𝐴 ∈ On → Ord 𝐴)
19 ordelsuc 7840 . . . . . . . . . . . . . 14 ((𝐵 ∈ ω ∧ Ord 𝐴) → (𝐵𝐴 ↔ suc 𝐵𝐴))
2018, 19sylan2 593 . . . . . . . . . . . . 13 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (𝐵𝐴 ↔ suc 𝐵𝐴))
2117, 20sylibd 239 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝐴 ∈ On) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
22 ssdomg 9040 . . . . . . . . . . . . 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 458 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
2625adantr 480 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴 → suc 𝐵𝐴))
2714, 26jcad 512 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴 → (𝐴 ≼ suc 𝐵 ∧ suc 𝐵𝐴)))
28 sbth 9133 . . . . . . . 8 ((𝐴 ≼ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐴 ≈ suc 𝐵)
2927, 28syl6 35 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (ω ⊆ 𝐴𝐴 ≈ suc 𝐵))
305, 29mtod 198 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → ¬ ω ⊆ 𝐴)
31 ordom 7897 . . . . . . . . 9 Ord ω
32 ordtri1 6417 . . . . . . . . 9 ((Ord ω ∧ Ord 𝐴) → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
3331, 18, 32sylancr 587 . . . . . . . 8 (𝐴 ∈ On → (ω ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ω))
3433con2bid 354 . . . . . . 7 (𝐴 ∈ On → (𝐴 ∈ ω ↔ ¬ ω ⊆ 𝐴))
3534ad2antrr 726 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 ∈ ω ↔ ¬ ω ⊆ 𝐴))
3630, 35mpbird 257 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 ∈ ω)
37 simplr 769 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐵 ∈ ω)
3836, 37jca 511 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → (𝐴 ∈ ω ∧ 𝐵 ∈ ω))
39 nneneq 9246 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
4039biimpa 476 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 = 𝐵)
4138, 40sylancom 588 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ ω) ∧ 𝐴𝐵) → 𝐴 = 𝐵)
4241ex 412 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
43 eqeng 9026 . . 3 (𝐴 ∈ On → (𝐴 = 𝐵𝐴𝐵))
4443adantr 480 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵𝐴𝐵))
4542, 44impbid 212 1 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wss 3951   class class class wbr 5143  Ord word 6383  Oncon0 6384  suc csuc 6386  ωcom 7887  cen 8982  cdom 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989
This theorem is referenced by: (None)
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