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Theorem onsupnub 42708
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 771 . 2 (((𝐴 ⊆ On ∧ 𝐴𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧𝐴 𝑧𝐵)) → ∀𝑧𝐴 𝑧𝐵)
2 unissb 4946 . 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 394  wcel 2098  wral 3058  wss 3949   cuni 4912  Oncon0 6374
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 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-v 3475  df-in 3956  df-ss 3966  df-uni 4913
This theorem is referenced by: (None)
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