Theorem onsupnub 43528

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 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)) → ∪ 𝐴 ⊆ 𝐵)

Distinct variable groups:   𝑧,𝐴   𝑧,𝐵

Allowed substitution hint:   𝑉(𝑧)

Proof of Theorem onsupnub

Step	Hyp	Ref	Expression
1		simprr 773	. 2 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)) → ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)
2		unissb 4895	. 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)
3	1, 2	sylibr 234	1 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)) → ∪ 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  ∀wral 3050   ⊆ wss 3900  ∪ cuni 4862  Oncon0 6316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-v 3441  df-ss 3917  df-uni 4863

This theorem is referenced by: (None)