Theorem eluni2f 45821

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 1891	. 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 45736	. 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))
5		df-rex 3090	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥))
6	1, 4, 5	3bitr4i 306	1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1809   ∈ wcel 2143  Ⅎwnfc 2910  ∃wrex 3089  ∪ cuni 4872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rex 3090  df-v 3457  df-uni 4873

This theorem is referenced by:  smfresal  47502  smfpimbor1lem2  47513