Theorem bj-prex 37184

Description: Existence of unordered pairs proved from ax-bj-sn 37177 and ax-bj-bun 37181. (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 4581	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snex 37179	. . 3 ⊢ {𝐴} ∈ V
3		bj-snex 37179	. . 3 ⊢ {𝐵} ∈ V
4		bj-unexg 37182	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	mp2an 692	. 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ V
6	1, 5	eqeltri 2830	1 ⊢ {𝐴, 𝐵} ∈ V

Syntax hints:   ∈ wcel 2113  Vcvv 3438   ∪ cun 3897  {csn 4578  {cpr 4580

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2182  ax-ext 2706  ax-nul 5249  ax-bj-sn 37177  ax-bj-bun 37181

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-dif 3902  df-un 3904  df-nul 4284  df-sn 4579  df-pr 4581

This theorem is referenced by: (None)