Theorem bj-prexg 37242

Description: Existence of unordered pairs formed on sets, proved from ax-bj-sn 37236 and ax-bj-bun 37240. Contrary to bj-prex 37243, this proof is intuitionistically valid and does not require ax-nul 5252. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-prexg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)

Proof of Theorem bj-prexg

Step	Hyp	Ref	Expression
1		df-pr 4584	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snexg 37237	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
3		bj-snexg 37237	. . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ∈ V)
4		bj-unexg 37241	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	syl2an 597	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ∪ {𝐵}) ∈ V)
6	1, 5	eqeltrid 2841	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114  Vcvv 3441   ∪ cun 3900  {csn 4581  {cpr 4583

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709  ax-bj-sn 37236  ax-bj-bun 37240

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-un 3907  df-sn 4582  df-pr 4584

This theorem is referenced by: (None)