Theorem bj-prexg 37183

Description: Existence of unordered pairs formed on sets, proved from ax-bj-sn 37177 and ax-bj-bun 37181. Contrary to bj-prex 37184, this proof is intuitionistically valid and does not require ax-nul 5249. (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 4581	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snexg 37178	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
3		bj-snexg 37178	. . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ∈ V)
4		bj-unexg 37182	. . 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 2838	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ 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-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-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-un 3904  df-sn 4579  df-pr 4581

This theorem is referenced by: (None)