Theorem unielid 42431

Description: Two ways to say the union of a class is an element of that class. (Contributed by RP, 27-Jan-2025.)

Assertion

Ref	Expression
unielid	⊢ (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem unielid

Step	Hyp	Ref	Expression
1		ssid 4004	. 2 ⊢ 𝐴 ⊆ 𝐴
2		unielss 42430	. 2 ⊢ (𝐴 ⊆ 𝐴 → (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥))
3	1, 2	ax-mp 5	1 ⊢ (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)

Syntax hints:   ↔ wb 205   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069   ⊆ wss 3948  ∪ cuni 4908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-v 3475  df-in 3955  df-ss 3965  df-uni 4909

This theorem is referenced by:  onsupnmax  42440  onsupeqmax  42458  onsupeqnmax  42459