Theorem bj-prex 37006

Description: Existence of unordered pairs proved from ax-bj-sn 36999 and ax-bj-bun 37003. (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 4651	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snex 37001	. . 3 ⊢ {𝐴} ∈ V
3		bj-snex 37001	. . 3 ⊢ {𝐵} ∈ V
4		bj-unexg 37004	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	mp2an 691	. 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ V
6	1, 5	eqeltri 2840	1 ⊢ {𝐴, 𝐵} ∈ V

Syntax hints:   ∈ wcel 2108  Vcvv 3488   ∪ cun 3974  {csn 4648  {cpr 4650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-nul 5324  ax-bj-sn 36999  ax-bj-bun 37003

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-un 3981  df-nul 4353  df-sn 4649  df-pr 4651

This theorem is referenced by: (None)