Theorem bj-prexg 37619

Description: Existence of unordered pairs formed on sets, proved from ax-bj-sn 37613 and ax-bj-bun 37617. Contrary to bj-prex 37620, this proof is intuitionistically valid and does not require ax-nul 5268. (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 4591	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snexg 37614	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
3		bj-snexg 37614	. . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ∈ V)
4		bj-unexg 37618	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	syl2an 607	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ∪ {𝐵}) ∈ V)
6	1, 5	eqeltrid 2865	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2141  Vcvv 3453   ∪ cun 3902  {csn 4588  {cpr 4590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-12 2211  ax-ext 2733  ax-bj-sn 37613  ax-bj-bun 37617

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3909  df-sn 4589  df-pr 4591

This theorem is referenced by: (None)