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Theorem bj-prexg 37040
Description: Existence of unordered pairs formed on sets, proved from ax-bj-sn 37034 and ax-bj-bun 37038. Contrary to bj-prex 37041, this proof is intuitionistically valid and does not require ax-nul 5306. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-prexg ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)

Proof of Theorem bj-prexg
StepHypRef Expression
1 df-pr 4629 . 2 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2 bj-snexg 37035 . . 3 (𝐴𝑉 → {𝐴} ∈ V)
3 bj-snexg 37035 . . 3 (𝐵𝑊 → {𝐵} ∈ V)
4 bj-unexg 37039 . . 3 (({𝐴} ∈ V ∧ {𝐵} ∈ V) → ({𝐴} ∪ {𝐵}) ∈ V)
52, 3, 4syl2an 596 . 2 ((𝐴𝑉𝐵𝑊) → ({𝐴} ∪ {𝐵}) ∈ V)
61, 5eqeltrid 2845 1 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  Vcvv 3480  cun 3949  {csn 4626  {cpr 4628
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 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-bj-sn 37034  ax-bj-bun 37038
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629
This theorem is referenced by: (None)
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