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Theorem onsupeqmax 42739
Description: Condition when the supremum of a set of ordinals is the maximum element of that set. (Contributed by RP, 24-Jan-2025.)
Assertion
Ref Expression
onsupeqmax ((𝐴 ⊆ On ∧ 𝐴𝑉) → (∃𝑥𝐴𝑦𝐴 𝑦𝑥 𝐴𝐴))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑉,𝑦

Proof of Theorem onsupeqmax
StepHypRef Expression
1 unielid 42712 . . 3 ( 𝐴𝐴 ↔ ∃𝑥𝐴𝑦𝐴 𝑦𝑥)
21a1i 11 . 2 ((𝐴 ⊆ On ∧ 𝐴𝑉) → ( 𝐴𝐴 ↔ ∃𝑥𝐴𝑦𝐴 𝑦𝑥))
32bicomd 222 1 ((𝐴 ⊆ On ∧ 𝐴𝑉) → (∃𝑥𝐴𝑦𝐴 𝑦𝑥 𝐴𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2098  wral 3051  wrex 3060  wss 3939   cuni 4903  Oncon0 6364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-ss 3956  df-uni 4904
This theorem is referenced by: (None)
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