Theorem difxp1ss 32554

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

Assertion

Ref	Expression
difxp1ss	⊢ ((𝐴 ∖ 𝐶) × 𝐵) ⊆ (𝐴 × 𝐵)

Proof of Theorem difxp1ss

Step	Hyp	Ref	Expression
1		difxp1 6198	. 2 ⊢ ((𝐴 ∖ 𝐶) × 𝐵) = ((𝐴 × 𝐵) ∖ (𝐶 × 𝐵))
2		difss 4159	. 2 ⊢ ((𝐴 × 𝐵) ∖ (𝐶 × 𝐵)) ⊆ (𝐴 × 𝐵)
3	1, 2	eqsstri 4043	1 ⊢ ((𝐴 ∖ 𝐶) × 𝐵) ⊆ (𝐴 × 𝐵)

Syntax hints:   ∖ cdif 3973   ⊆ wss 3976   × cxp 5698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708

This theorem is referenced by: (None)