Theorem bj-prex 37406

Description: Existence of unordered pairs proved from ax-bj-sn 37399 and ax-bj-bun 37403. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-prex	⊢ {𝐴, 𝐵} ∈ V

Proof of Theorem bj-prex

Step	Hyp	Ref	Expression
1		df-pr 4560	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snex 37401	. . 3 ⊢ {𝐴} ∈ V
3		bj-snex 37401	. . 3 ⊢ {𝐵} ∈ V
4		bj-unexg 37404	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	mp2an 699	. 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ V
6	1, 5	eqeltri 2837	1 ⊢ {𝐴, 𝐵} ∈ V

Syntax hints:   ∈ wcel 2121  Vcvv 3433   ∪ cun 3882  {csn 4557  {cpr 4559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-12 2191  ax-ext 2713  ax-nul 5230  ax-bj-sn 37399  ax-bj-bun 37403

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-dif 3887  df-un 3889  df-nul 4264  df-sn 4558  df-pr 4560

This theorem is referenced by: (None)