Theorem bj-prex 36225

Description: Existence of unordered pairs proved from ax-bj-sn 36218 and ax-bj-bun 36222. (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 4631	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snex 36220	. . 3 ⊢ {𝐴} ∈ V
3		bj-snex 36220	. . 3 ⊢ {𝐵} ∈ V
4		bj-unexg 36223	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	mp2an 689	. 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ V
6	1, 5	eqeltri 2828	1 ⊢ {𝐴, 𝐵} ∈ V

Syntax hints:   ∈ wcel 2105  Vcvv 3473   ∪ cun 3946  {csn 4628  {cpr 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702  ax-nul 5306  ax-bj-sn 36218  ax-bj-bun 36222

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-dif 3951  df-un 3953  df-nul 4323  df-sn 4629  df-pr 4631

This theorem is referenced by: (None)