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Theorem bj-unexg 37021
Description: Existence of binary unions of sets, proved from ax-bj-bun 37020. (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-unexg ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Proof of Theorem bj-unexg
Dummy variables 𝑥 𝑦 𝑧 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elissetv 2820 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elissetv 2820 . 2 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
3 exdistrv 1953 . . 3 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
4 uneq12 4173 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦) = (𝐴𝐵))
5 ax-bj-bun 37020 . . . . . . . . 9 𝑥𝑦𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡𝑦))
65spi 2182 . . . . . . . 8 𝑦𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡𝑦))
76spi 2182 . . . . . . 7 𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡𝑦))
8 bj-axbun 37019 . . . . . . 7 ((𝑥𝑦) ∈ V ↔ ∃𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡𝑦)))
97, 8mpbir 231 . . . . . 6 (𝑥𝑦) ∈ V
104, 9eqeltrrdi 2848 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝐴𝐵) ∈ V)
1110exlimiv 1928 . . . 4 (∃𝑦(𝑥 = 𝐴𝑦 = 𝐵) → (𝐴𝐵) ∈ V)
1211exlimiv 1928 . . 3 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → (𝐴𝐵) ∈ V)
133, 12sylbir 235 . 2 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → (𝐴𝐵) ∈ V)
141, 2, 13syl2an 596 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  wal 1535   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  cun 3961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-bj-bun 37020
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968
This theorem is referenced by:  bj-prexg  37022  bj-prex  37023  bj-adjfrombun  37029
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