Theorem onsupnub 42914

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 771	. 2 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)) → ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)
2		unissb 4947	. 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)
3	1, 2	sylibr 233	1 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)) → ∪ 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2099  ∀wral 3051   ⊆ wss 3947  ∪ cuni 4913  Oncon0 6376

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-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-v 3464  df-ss 3964  df-uni 4914

This theorem is referenced by: (None)