Theorem bj-prex 37875

Description: Existence of unordered pairs proved from ax-bj-sn 37868 and ax-bj-bun 37872. (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 4586	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snex 37870	. . 3 ⊢ {𝐴} ∈ V
3		bj-snex 37870	. . 3 ⊢ {𝐵} ∈ V
4		bj-unexg 37873	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	mp2an 705	. 2 ⊢ ({𝐴} ∪ {𝐵}) ∈ V
6	1, 5	eqeltri 2856	1 ⊢ {𝐴, 𝐵} ∈ V

This proof depends on syntax axioms:   ∈ wcel 2145  Vcvv 3450   ∪ cun 3896  {csn 4583  {cpr 4585

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2732  ax-nul 5259  ax-bj-sn 37868  ax-bj-bun 37872

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 3901  df-un 3903  df-nul 4279  df-sn 4584  df-pr 4586

This theorem is used by: (None)