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

Theorem onsupnub 42571
Description: An upper bound of a set of ordinals is not less than the supremum. (Contributed by RP, 27-Jan-2025.)
Assertion
Ref Expression
onsupnub (((𝐴 ⊆ On ∧ 𝐴𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧𝐴 𝑧𝐵)) → 𝐴𝐵)
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵
Allowed substitution hint:   𝑉(𝑧)

Proof of Theorem onsupnub
StepHypRef Expression
1 simprr 770 . 2 (((𝐴 ⊆ On ∧ 𝐴𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧𝐴 𝑧𝐵)) → ∀𝑧𝐴 𝑧𝐵)
2 unissb 4936 . 2 ( 𝐴𝐵 ↔ ∀𝑧𝐴 𝑧𝐵)
31, 2sylibr 233 1 (((𝐴 ⊆ On ∧ 𝐴𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧𝐴 𝑧𝐵)) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  wral 3055  wss 3943   cuni 4902  Oncon0 6358
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-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-v 3470  df-in 3950  df-ss 3960  df-uni 4903
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator