Theorem unielid 43664

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 3937	. 2 ⊢ 𝐴 ⊆ 𝐴
2		unielss 43663	. 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 207   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   ⊆ wss 3883  ∪ cuni 4838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-ss 3900  df-uni 4839

This theorem is referenced by:  onsupnmax  43673  onsupeqmax  43691  onsupeqnmax  43692