Theorem bj-prex 37023

Description: Existence of unordered pairs proved from ax-bj-sn 37016 and ax-bj-bun 37020. (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 4634	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snex 37018	. . 3 ⊢ {𝐴} ∈ V
3		bj-snex 37018	. . 3 ⊢ {𝐵} ∈ V
4		bj-unexg 37021	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	mp2an 692	. 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ V
6	1, 5	eqeltri 2835	1 ⊢ {𝐴, 𝐵} ∈ V

Syntax hints:   ∈ wcel 2106  Vcvv 3478   ∪ cun 3961  {csn 4631  {cpr 4633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-nul 5312  ax-bj-sn 37016  ax-bj-bun 37020

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-nul 4340  df-sn 4632  df-pr 4634

This theorem is referenced by: (None)