Theorem bj-prex 37335

Description: Existence of unordered pairs proved from ax-bj-sn 37328 and ax-bj-bun 37332. (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 37330	. . 3 ⊢ {𝐴} ∈ V
3		bj-snex 37330	. . 3 ⊢ {𝐵} ∈ V
4		bj-unexg 37333	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	mp2an 693	. 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ V
6	1, 5	eqeltri 2831	1 ⊢ {𝐴, 𝐵} ∈ V

Syntax hints:   ∈ wcel 2114  Vcvv 3427   ∪ cun 3883  {csn 4557  {cpr 4559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2184  ax-ext 2707  ax-nul 5230  ax-bj-sn 37328  ax-bj-bun 37332

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-v 3429  df-dif 3888  df-un 3890  df-nul 4264  df-sn 4558  df-pr 4560

This theorem is referenced by: (None)