Theorem unielid 43222

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 4019	. 2 ⊢ 𝐴 ⊆ 𝐴
2		unielss 43221	. 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 2107  ∀wral 3060  ∃wrex 3069   ⊆ wss 3964  ∪ cuni 4913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1541  df-ex 1778  df-nf 1782  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-ss 3981  df-uni 4914

This theorem is referenced by:  onsupnmax  43231  onsupeqmax  43249  onsupeqnmax  43250