Theorem difxp1ss 32543

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 6195	. 2 ⊢ ((𝐴 ∖ 𝐶) × 𝐵) = ((𝐴 × 𝐵) ∖ (𝐶 × 𝐵))
2		difss 4153	. 2 ⊢ ((𝐴 × 𝐵) ∖ (𝐶 × 𝐵)) ⊆ (𝐴 × 𝐵)
3	1, 2	eqsstri 4037	1 ⊢ ((𝐴 ∖ 𝐶) × 𝐵) ⊆ (𝐴 × 𝐵)

Syntax hints:   ∖ cdif 3967   ⊆ wss 3970   × cxp 5697

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 2105  ax-9 2113  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450

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 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-xp 5705  df-rel 5706  df-cnv 5707

This theorem is referenced by: (None)