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Theorem onmaxnelsup 41957
Description: Two ways to say the maximum element of a class of ordinals is also the supremum of that class. (Contributed by RP, 27-Jan-2025.)
Assertion
Ref Expression
onmaxnelsup (𝐴 ⊆ On → (¬ 𝐴 𝐴 ↔ ∃𝑥𝐴𝑦𝐴 𝑦𝑥))
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem onmaxnelsup
StepHypRef Expression
1 rexnal 3100 . . 3 (∃𝑥𝐴 ¬ ∃𝑦𝐴 𝑥𝑦 ↔ ¬ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)
2 ralnex 3072 . . . 4 (∀𝑦𝐴 ¬ 𝑥𝑦 ↔ ¬ ∃𝑦𝐴 𝑥𝑦)
32rexbii 3094 . . 3 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦 ↔ ∃𝑥𝐴 ¬ ∃𝑦𝐴 𝑥𝑦)
4 ssunib 41954 . . . 4 (𝐴 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)
54notbii 319 . . 3 𝐴 𝐴 ↔ ¬ ∀𝑥𝐴𝑦𝐴 𝑥𝑦)
61, 3, 53bitr4ri 303 . 2 𝐴 𝐴 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
7 simpl 483 . . . . . 6 ((𝐴 ⊆ On ∧ 𝑥𝐴) → 𝐴 ⊆ On)
87sselda 3981 . . . . 5 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑦 ∈ On)
9 ssel2 3976 . . . . . 6 ((𝐴 ⊆ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
109adantr 481 . . . . 5 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ 𝑦𝐴) → 𝑥 ∈ On)
11 ontri1 6395 . . . . 5 ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦𝑥 ↔ ¬ 𝑥𝑦))
128, 10, 11syl2anc 584 . . . 4 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ 𝑦𝐴) → (𝑦𝑥 ↔ ¬ 𝑥𝑦))
1312ralbidva 3175 . . 3 ((𝐴 ⊆ On ∧ 𝑥𝐴) → (∀𝑦𝐴 𝑦𝑥 ↔ ∀𝑦𝐴 ¬ 𝑥𝑦))
1413rexbidva 3176 . 2 (𝐴 ⊆ On → (∃𝑥𝐴𝑦𝐴 𝑦𝑥 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦))
156, 14bitr4id 289 1 (𝐴 ⊆ On → (¬ 𝐴 𝐴 ↔ ∃𝑥𝐴𝑦𝐴 𝑦𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wcel 2106  wral 3061  wrex 3070  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 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  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
This theorem is referenced by: (None)
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