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Theorem difxp1ss 30385
 Description: Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.)
Assertion
Ref Expression
difxp1ss ((𝐴𝐶) × 𝐵) ⊆ (𝐴 × 𝐵)

Proof of Theorem difxp1ss
StepHypRef Expression
1 difxp1 5995 . 2 ((𝐴𝐶) × 𝐵) = ((𝐴 × 𝐵) ∖ (𝐶 × 𝐵))
2 difss 4038 . 2 ((𝐴 × 𝐵) ∖ (𝐶 × 𝐵)) ⊆ (𝐴 × 𝐵)
31, 2eqsstri 3927 1 ((𝐴𝐶) × 𝐵) ⊆ (𝐴 × 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∖ cdif 3856   ⊆ wss 3859   × cxp 5523 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-xp 5531  df-rel 5532  df-cnv 5533 This theorem is referenced by: (None)
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