Theorem onsupeqmax 42674

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 42647	. . 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 222	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∪ 𝐴 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2099  ∀wral 3058  ∃wrex 3067   ⊆ wss 3947  ∪ cuni 4908  Oncon0 6369

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-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-v 3473  df-in 3954  df-ss 3964  df-uni 4909

This theorem is referenced by: (None)