Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-unexg Structured version   Visualization version   GIF version

Theorem bj-unexg 37039
Description: Existence of binary unions of sets, proved from ax-bj-bun 37038. (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 2822 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elissetv 2822 . 2 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
3 exdistrv 1955 . . 3 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
4 uneq12 4163 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑦) = (𝐴𝐵))
5 ax-bj-bun 37038 . . . . . . . . 9 𝑥𝑦𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡𝑦))
65spi 2184 . . . . . . . 8 𝑦𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡𝑦))
76spi 2184 . . . . . . 7 𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡𝑦))
8 bj-axbun 37037 . . . . . . 7 ((𝑥𝑦) ∈ V ↔ ∃𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡𝑦)))
97, 8mpbir 231 . . . . . 6 (𝑥𝑦) ∈ V
104, 9eqeltrrdi 2850 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝐴𝐵) ∈ V)
1110exlimiv 1930 . . . 4 (∃𝑦(𝑥 = 𝐴𝑦 = 𝐵) → (𝐴𝐵) ∈ V)
1211exlimiv 1930 . . 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 848  wal 1538   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cun 3949
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-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
This theorem is referenced by:  bj-prexg  37040  bj-prex  37041  bj-adjfrombun  37047
  Copyright terms: Public domain W3C validator