Theorem bj-prex 37393

Description: Existence of unordered pairs proved from ax-bj-sn 37386 and ax-bj-bun 37390. (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 4558	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snex 37388	. . 3 ⊢ {𝐴} ∈ V
3		bj-snex 37388	. . 3 ⊢ {𝐵} ∈ V
4		bj-unexg 37391	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	mp2an 698	. 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ V
6	1, 5	eqeltri 2835	1 ⊢ {𝐴, 𝐵} ∈ V

Syntax hints:   ∈ wcel 2119  Vcvv 3431   ∪ cun 3881  {csn 4555  {cpr 4557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2711  ax-nul 5228  ax-bj-sn 37386  ax-bj-bun 37390

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-dif 3886  df-un 3888  df-nul 4262  df-sn 4556  df-pr 4558

This theorem is referenced by: (None)