Theorem onsupeqmax 43823

Description: Condition when the supremum of a set of ordinals is the maximum element of that set. (Contributed by RP, 24-Jan-2025.)

Assertion

Ref	Expression
onsupeqmax	⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∪ 𝐴 ∈ 𝐴))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑉,𝑦

Proof of Theorem onsupeqmax

Step	Hyp	Ref	Expression
1		unielid 43796	. . 3 ⊢ (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
2	1	a1i 11	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥))
3	2	bicomd 225	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∪ 𝐴 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086   ⊆ wss 3904  ∪ cuni 4865  Oncon0 6346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-ss 3921  df-uni 4866

This theorem is referenced by: (None)