Theorem eluni2f 45678

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 1881	. 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 45593	. 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))
5		df-rex 3087	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥))
6	1, 4, 5	3bitr4i 305	1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 208   ∧ wa 399  ∃wex 1799   ∈ wcel 2142  Ⅎwnfc 2909  ∃wrex 3086  ∪ cuni 4865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rex 3087  df-v 3456  df-uni 4866

This theorem is referenced by:  smfresal  47359  smfpimbor1lem2  47370