Theorem bj-prex 37041

Description: Existence of unordered pairs proved from ax-bj-sn 37034 and ax-bj-bun 37038. (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 4629	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snex 37036	. . 3 ⊢ {𝐴} ∈ V
3		bj-snex 37036	. . 3 ⊢ {𝐵} ∈ V
4		bj-unexg 37039	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	mp2an 692	. 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ V
6	1, 5	eqeltri 2837	1 ⊢ {𝐴, 𝐵} ∈ V

Syntax hints:   ∈ wcel 2108  Vcvv 3480   ∪ cun 3949  {csn 4626  {cpr 4628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-nul 5306  ax-bj-sn 37034  ax-bj-bun 37038

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-pr 4629

This theorem is referenced by: (None)