Theorem difxp2ss 30850

Description: Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.)

Assertion

Ref	Expression
difxp2ss	⊢ (𝐴 × (𝐵 ∖ 𝐶)) ⊆ (𝐴 × 𝐵)

Proof of Theorem difxp2ss

Step	Hyp	Ref	Expression
1		difxp2 6066	. 2 ⊢ (𝐴 × (𝐵 ∖ 𝐶)) = ((𝐴 × 𝐵) ∖ (𝐴 × 𝐶))
2		difss 4070	. 2 ⊢ ((𝐴 × 𝐵) ∖ (𝐴 × 𝐶)) ⊆ (𝐴 × 𝐵)
3	1, 2	eqsstri 3959	1 ⊢ (𝐴 × (𝐵 ∖ 𝐶)) ⊆ (𝐴 × 𝐵)

Syntax hints:   ∖ cdif 3888   ⊆ wss 3891   × cxp 5586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-opab 5141  df-xp 5594  df-rel 5595

This theorem is referenced by: (None)