Theorem bj-prex 37704

Description: Existence of unordered pairs proved from ax-bj-sn 37697 and ax-bj-bun 37701. (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 4591	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snex 37699	. . 3 ⊢ {𝐴} ∈ V
3		bj-snex 37699	. . 3 ⊢ {𝐵} ∈ V
4		bj-unexg 37702	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	mp2an 704	. 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ V
6	1, 5	eqeltri 2858	1 ⊢ {𝐴, 𝐵} ∈ V

This proof depends on syntax axioms:   ∈ wcel 2142  Vcvv 3454   ∪ cun 3902  {csn 4588  {cpr 4590

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-12 2212  ax-ext 2734  ax-nul 5268  ax-bj-sn 37697  ax-bj-bun 37701

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-dif 3907  df-un 3909  df-nul 4286  df-sn 4589  df-pr 4591

This theorem is used by: (None)