Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xpexd Structured version   Visualization version   GIF version

Theorem xpexd 38314
 Description: The Cartesian product of two sets is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
xpexd.1 (𝜑𝐴𝑉)
xpexd.2 (𝜑𝐵𝑊)
Assertion
Ref Expression
xpexd (𝜑 → (𝐴 × 𝐵) ∈ V)

Proof of Theorem xpexd
StepHypRef Expression
1 xpexd.1 . 2 (𝜑𝐴𝑉)
2 xpexd.2 . 2 (𝜑𝐵𝑊)
3 xpexg 6858 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
41, 2, 3syl2anc 691 1 (𝜑 → (𝐴 × 𝐵) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  Vcvv 3173   × cxp 5036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-opab 4644  df-xp 5044  df-rel 5045 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator