Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  onsupeqmax Structured version   Visualization version   GIF version

Theorem onsupeqmax 42674
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 42647 . . 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 395  wcel 2099  wral 3058  wrex 3067  wss 3947   cuni 4908  Oncon0 6369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-v 3473  df-in 3954  df-ss 3964  df-uni 4909
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator