Theorem unielid 43576

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 3958	. 2 ⊢ 𝐴 ⊆ 𝐴
2		unielss 43575	. 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 206   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3903  ∪ cuni 4865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-ss 3920  df-uni 4866

This theorem is referenced by:  onsupnmax  43585  onsupeqmax  43603  onsupeqnmax  43604