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Theorem onsupeqnmax 41929
Description: Condition when the supremum of a class of ordinals is not the maximum element of that class. (Contributed by RP, 27-Jan-2025.)
Assertion
Ref Expression
onsupeqnmax (𝐴 ⊆ On → (∀𝑥𝐴𝑦𝐴 𝑥𝑦 ↔ ( 𝐴 = 𝐴 ∧ ¬ 𝐴𝐴)))
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem onsupeqnmax
StepHypRef Expression
1 simpl 484 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝑥𝐴) → 𝐴 ⊆ On)
21sselda 3981 . . . . . . . 8 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑦 ∈ On)
3 ssel2 3976 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
43adantr 482 . . . . . . . 8 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑥 ∈ On)
5 ontri1 6395 . . . . . . . 8 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ ¬ 𝑥𝑦))
62, 4, 5syl2anc 585 . . . . . . 7 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ 𝑦𝐴) → (𝑦𝑥 ↔ ¬ 𝑥𝑦))
76ralbidva 3176 . . . . . 6 ((𝐴 ⊆ On ∧ 𝑥𝐴) → (∀𝑦𝐴 𝑦𝑥 ↔ ∀𝑦𝐴 ¬ 𝑥𝑦))
87rexbidva 3177 . . . . 5 (𝐴 ⊆ On → (∃𝑥𝐴𝑦𝐴 𝑦𝑥 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦))
98notbid 318 . . . 4 (𝐴 ⊆ On → (¬ ∃𝑥𝐴𝑦𝐴 𝑦𝑥 ↔ ¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦))
109bicomd 222 . . 3 (𝐴 ⊆ On → (¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ↔ ¬ ∃𝑥𝐴𝑦𝐴 𝑦𝑥))
11 dfrex2 3074 . . . . 5 (∃𝑦𝐴 𝑥𝑦 ↔ ¬ ∀𝑦𝐴 ¬ 𝑥𝑦)
1211ralbii 3094 . . . 4 (∀𝑥𝐴𝑦𝐴 𝑥𝑦 ↔ ∀𝑥𝐴 ¬ ∀𝑦𝐴 ¬ 𝑥𝑦)
13 ralnex 3073 . . . 4 (∀𝑥𝐴 ¬ ∀𝑦𝐴 ¬ 𝑥𝑦 ↔ ¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
1412, 13bitri 275 . . 3 (∀𝑥𝐴𝑦𝐴 𝑥𝑦 ↔ ¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
15 unielid 41901 . . . 4 ( 𝐴𝐴 ↔ ∃𝑥𝐴𝑦𝐴 𝑦𝑥)
1615notbii 320 . . 3 𝐴𝐴 ↔ ¬ ∃𝑥𝐴𝑦𝐴 𝑦𝑥)
1710, 14, 163bitr4g 314 . 2 (𝐴 ⊆ On → (∀𝑥𝐴𝑦𝐴 𝑥𝑦 ↔ ¬ 𝐴𝐴))
18 onsupnmax 41910 . . 3 (𝐴 ⊆ On → (¬ 𝐴𝐴 𝐴 = 𝐴))
1918pm4.71rd 564 . 2 (𝐴 ⊆ On → (¬ 𝐴𝐴 ↔ ( 𝐴 = 𝐴 ∧ ¬ 𝐴𝐴)))
2017, 19bitrd 279 1 (𝐴 ⊆ On → (∀𝑥𝐴𝑦𝐴 𝑥𝑦 ↔ ( 𝐴 = 𝐴 ∧ ¬ 𝐴𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  wss 3947   cuni 4907  Oncon0 6361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365  df-suc 6367
This theorem is referenced by: (None)
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