Theorem bj-prex 35711

Description: Existence of unordered pairs proved from ax-bj-sn 35704 and ax-bj-bun 35708. (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 4624	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snex 35706	. . 3 ⊢ {𝐴} ∈ V
3		bj-snex 35706	. . 3 ⊢ {𝐵} ∈ V
4		bj-unexg 35709	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	mp2an 690	. 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ V
6	1, 5	eqeltri 2828	1 ⊢ {𝐴, 𝐵} ∈ V

Syntax hints:   ∈ wcel 2106  Vcvv 3472   ∪ cun 3941  {csn 4621  {cpr 4623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2702  ax-nul 5298  ax-bj-sn 35704  ax-bj-bun 35708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3474  df-dif 3946  df-un 3948  df-nul 4318  df-sn 4622  df-pr 4624

This theorem is referenced by: (None)