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Theorem xpexd 38767
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 6913 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
41, 2, 3syl2anc 692 1 (𝜑 → (𝐴 × 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  Vcvv 3186   × cxp 5072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-opab 4674  df-xp 5080  df-rel 5081
This theorem is referenced by: (None)
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