Theorem onsupnub 43946

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 784	. 2 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)) → ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)
2		unissb 4905	. 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)
3	1, 2	sylibr 237	1 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)) → ∪ 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2141  ∀wral 3077   ⊆ wss 3904  ∪ cuni 4871  Oncon0 6360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-v 3455  df-ss 3921  df-uni 4872

This theorem is referenced by: (None)