Theorem onsupnub 43210

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 772	. 2 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ On ∧ ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)) → ∀𝑧 ∈ 𝐴 𝑧 ⊆ 𝐵)
2		unissb 4911	. 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 2109  ∀wral 3046   ⊆ wss 3922  ∪ cuni 4879  Oncon0 6340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3047  df-v 3457  df-ss 3939  df-uni 4880

This theorem is referenced by: (None)