Theorem eluni2f 39772

Description: Membership in class union. Restricted quantifier version. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
eluni2f.1	⊢ Ⅎ𝑥𝐴
eluni2f.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
eluni2f	⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)

Distinct variable group:   𝐴,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem eluni2f

Step	Hyp	Ref	Expression
1		exancom 1947	. 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥))
2		eluni2f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		eluni2f.2	. . 3 ⊢ Ⅎ𝑥𝐵
4	2, 3	elunif 39663	. 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))
5		df-rex 3098	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥))
6	1, 4, 5	3bitr4i 294	1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 197   ∧ wa 384  ∃wex 1859   ∈ wcel 2155  Ⅎwnfc 2931  ∃wrex 3093  ∪ cuni 4623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-rex 3098  df-v 3389  df-uni 4624

This theorem is referenced by:  smfresal  41471  smfpimbor1lem2  41482