Theorem uniclaxun 45445

Description: A class that is closed under the union operation models the Axiom of Union ax-un 7682. Lemma II.2.4(5) of [Kunen2] p. 111. (Contributed by Eric Schmidt, 1-Oct-2025.)

Assertion

Ref	Expression
uniclaxun	⊢ (∀𝑥 ∈ 𝑀 ∪ 𝑥 ∈ 𝑀 → ∀𝑥 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∃𝑤 ∈ 𝑀 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝑦,𝑀

Allowed substitution hints:   𝑀(𝑥,𝑧,𝑤)

Proof of Theorem uniclaxun

Step	Hyp	Ref	Expression
1		rexex 3071	. . . . 5 ⊢ (∃𝑤 ∈ 𝑀 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))
2		eluni 4844	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))
3	1, 2	sylibr 236	. . . 4 ⊢ (∃𝑤 ∈ 𝑀 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ ∪ 𝑥)
4	3	rgenw 3059	. . 3 ⊢ ∀𝑧 ∈ 𝑀 (∃𝑤 ∈ 𝑀 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ ∪ 𝑥)
5		eleq2 2830	. . . . . 6 ⊢ (𝑦 = ∪ 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥))
6	5	imbi2d 342	. . . . 5 ⊢ (𝑦 = ∪ 𝑥 → ((∃𝑤 ∈ 𝑀 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ (∃𝑤 ∈ 𝑀 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ ∪ 𝑥)))
7	6	ralbidv 3164	. . . 4 ⊢ (𝑦 = ∪ 𝑥 → (∀𝑧 ∈ 𝑀 (∃𝑤 ∈ 𝑀 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ∀𝑧 ∈ 𝑀 (∃𝑤 ∈ 𝑀 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ ∪ 𝑥)))
8	7	rspcev 3562	. . 3 ⊢ ((∪ 𝑥 ∈ 𝑀 ∧ ∀𝑧 ∈ 𝑀 (∃𝑤 ∈ 𝑀 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ ∪ 𝑥)) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∃𝑤 ∈ 𝑀 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
9	4, 8	mpan2 698	. 2 ⊢ (∪ 𝑥 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∃𝑤 ∈ 𝑀 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
10	9	ralimi 3078	1 ⊢ (∀𝑥 ∈ 𝑀 ∪ 𝑥 ∈ 𝑀 → ∀𝑥 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∃𝑤 ∈ 𝑀 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548  ∃wex 1787   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065  ∪ cuni 4841

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-v 3435  df-uni 4842

This theorem is referenced by:  wfaxun  45458