Theorem eluni2f 40909

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 1842	. 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 40812	. 2 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))
5		df-rex 3111	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴 ∈ 𝑥))
6	1, 4, 5	3bitr4i 304	1 ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 207   ∧ wa 396  ∃wex 1761   ∈ wcel 2081  Ⅎwnfc 2933  ∃wrex 3106  ∪ cuni 4745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rex 3111  df-v 3439  df-uni 4746

This theorem is referenced by:  smfresal  42605  smfpimbor1lem2  42616