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Theorem difxp2ss 32399
Description: Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.)
Assertion
Ref Expression
difxp2ss (𝐴 × (𝐵𝐶)) ⊆ (𝐴 × 𝐵)

Proof of Theorem difxp2ss
StepHypRef Expression
1 difxp2 6172 . 2 (𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∖ (𝐴 × 𝐶))
2 difss 4128 . 2 ((𝐴 × 𝐵) ∖ (𝐴 × 𝐶)) ⊆ (𝐴 × 𝐵)
31, 2eqsstri 4011 1 (𝐴 × (𝐵𝐶)) ⊆ (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  cdif 3941  wss 3944   × cxp 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-opab 5212  df-xp 5684  df-rel 5685
This theorem is referenced by: (None)
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