Theorem difxp2ss 32566

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 6194	. 2 ⊢ (𝐴 × (𝐵 ∖ 𝐶)) = ((𝐴 × 𝐵) ∖ (𝐴 × 𝐶))
2		difss 4149	. 2 ⊢ ((𝐴 × 𝐵) ∖ (𝐴 × 𝐶)) ⊆ (𝐴 × 𝐵)
3	1, 2	eqsstri 4033	1 ⊢ (𝐴 × (𝐵 ∖ 𝐶)) ⊆ (𝐴 × 𝐵)

Syntax hints:   ∖ cdif 3963   ⊆ wss 3966   × cxp 5691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-opab 5214  df-xp 5699  df-rel 5700

This theorem is referenced by: (None)