Theorem unielid 43403

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 3954	. 2 ⊢ 𝐴 ⊆ 𝐴
2		unielss 43402	. 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 2113  ∀wral 3049  ∃wrex 3058   ⊆ wss 3899  ∪ cuni 4861

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-ss 3916  df-uni 4862

This theorem is referenced by:  onsupnmax  43412  onsupeqmax  43430  onsupeqnmax  43431