Theorem bj-prexg 37062

Description: Existence of unordered pairs formed on sets, proved from ax-bj-sn 37056 and ax-bj-bun 37060. Contrary to bj-prex 37063, this proof is intuitionistically valid and does not require ax-nul 5281. (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 4609	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snexg 37057	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
3		bj-snexg 37057	. . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ∈ V)
4		bj-unexg 37061	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	syl2an 596	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ∪ {𝐵}) ∈ V)
6	1, 5	eqeltrid 2839	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Vcvv 3464   ∪ cun 3929  {csn 4606  {cpr 4608

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 2708  ax-bj-sn 37056  ax-bj-bun 37060

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-un 3936  df-sn 4607  df-pr 4609

This theorem is referenced by: (None)