Theorem eluni2f 42542

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 1865	. 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 42448	. 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))
5		df-rex 3069	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥))
6	1, 4, 5	3bitr4i 302	1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 205   ∧ wa 395  ∃wex 1783   ∈ wcel 2108  Ⅎwnfc 2886  ∃wrex 3064  ∪ cuni 4836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rex 3069  df-v 3424  df-uni 4837

This theorem is referenced by:  smfresal  44209  smfpimbor1lem2  44220