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Theorem onsupnub 43151
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 772 . 2 (((𝐴 ⊆ On ∧ 𝐴𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧𝐴 𝑧𝐵)) → ∀𝑧𝐴 𝑧𝐵)
2 unissb 4965 . 2 ( 𝐴𝐵 ↔ ∀𝑧𝐴 𝑧𝐵)
31, 2sylibr 234 1 (((𝐴 ⊆ On ∧ 𝐴𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧𝐴 𝑧𝐵)) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2103  wral 3063  wss 3970   cuni 4931  Oncon0 6394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-v 3484  df-ss 3987  df-uni 4932
This theorem is referenced by: (None)
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