Theorem onsuplub 44003

Description: The supremum of a set of ordinals is the least upper bound. (Contributed by RP, 27-Jan-2025.)

Assertion

Ref	Expression
onsuplub	⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∈ On) → (𝐵 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝐵 ∈ 𝑧))

Distinct variable groups:   𝑧,𝐴   𝑧,𝐵

Allowed substitution hint:   𝑉(𝑧)

Proof of Theorem onsuplub

Step	Hyp	Ref	Expression
1		eluni2 4875	. 2 ⊢ (𝐵 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝐵 ∈ 𝑧)
2	1	a1i 11	1 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∈ On) → (𝐵 ∈ ∪ 𝐴 ↔ ∃𝑧 ∈ 𝐴 𝐵 ∈ 𝑧))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2142  ∃wrex 3088   ⊆ wss 3904  ∪ cuni 4871  Oncon0 6360

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3089  df-v 3456  df-uni 4872

This theorem is used by: (None)