Theorem bj-prexg 37005

Description: Existence of unordered pairs formed on sets, proved from ax-bj-sn 36999 and ax-bj-bun 37003. Contrary to bj-prex 37006, this proof is intuitionistically valid and does not require ax-nul 5324. (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 4651	. 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2		bj-snexg 37000	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V)
3		bj-snexg 37000	. . 3 ⊢ (𝐵 ∈ 𝑊 → {𝐵} ∈ V)
4		bj-unexg 37004	. . 3 ⊢ (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
5	2, 3, 4	syl2an 595	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ∪ {𝐵}) ∈ V)
6	1, 5	eqeltrid 2848	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  Vcvv 3488   ∪ cun 3974  {csn 4648  {cpr 4650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-bj-sn 36999  ax-bj-bun 37003

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-un 3981  df-sn 4649  df-pr 4651

This theorem is referenced by: (None)