Theorem bj-prex 37035

Description: Existence of unordered pairs proved from ax-bj-sn 37028 and ax-bj-bun 37032. (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 4595	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snex 37030	. . 3 ⊢ {𝐴} ∈ V
3		bj-snex 37030	. . 3 ⊢ {𝐵} ∈ V
4		bj-unexg 37033	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	mp2an 692	. 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ V
6	1, 5	eqeltri 2825	1 ⊢ {𝐴, 𝐵} ∈ V

Syntax hints:   ∈ wcel 2109  Vcvv 3450   ∪ cun 3915  {csn 4592  {cpr 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2702  ax-nul 5264  ax-bj-sn 37028  ax-bj-bun 37032

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3920  df-un 3922  df-nul 4300  df-sn 4593  df-pr 4595

This theorem is referenced by: (None)