Theorem bj-prex 37995

Description: Existence of unordered pairs proved from ax-bj-sn 37988 and ax-bj-bun 37992. (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 4586	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snex 37990	. . 3 ⊢ {𝐴} ∈ V
3		bj-snex 37990	. . 3 ⊢ {𝐵} ∈ V
4		bj-unexg 37993	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	mp2an 705	. 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ V
6	1, 5	eqeltri 2856	1 ⊢ {𝐴, 𝐵} ∈ V

This proof depends on syntax axioms:   ∈ wcel 2144  Vcvv 3450   ∪ cun 4149  {csn 4583  {cpr 4585

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732  ax-nul 5259  ax-bj-sn 37988  ax-bj-bun 37992

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 4047  df-un 4151  df-nul 4279  df-sn 4584  df-pr 4586

This theorem is used by: (None)