Theorem bj-prex 37028

Description: Existence of unordered pairs proved from ax-bj-sn 37021 and ax-bj-bun 37025. (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 4592	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snex 37023	. . 3 ⊢ {𝐴} ∈ V
3		bj-snex 37023	. . 3 ⊢ {𝐵} ∈ V
4		bj-unexg 37026	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	mp2an 692	. 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ V
6	1, 5	eqeltri 2824	1 ⊢ {𝐴, 𝐵} ∈ V

Syntax hints:   ∈ wcel 2109  Vcvv 3447   ∪ cun 3912  {csn 4589  {cpr 4591

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 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2701  ax-nul 5261  ax-bj-sn 37021  ax-bj-bun 37025

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-dif 3917  df-un 3919  df-nul 4297  df-sn 4590  df-pr 4592

This theorem is referenced by: (None)