Theorem difxp1ss 31621

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 6152	. 2 ⊢ ((𝐴 ∖ 𝐶) × 𝐵) = ((𝐴 × 𝐵) ∖ (𝐶 × 𝐵))
2		difss 4126	. 2 ⊢ ((𝐴 × 𝐵) ∖ (𝐶 × 𝐵)) ⊆ (𝐴 × 𝐵)
3	1, 2	eqsstri 4011	1 ⊢ ((𝐴 ∖ 𝐶) × 𝐵) ⊆ (𝐴 × 𝐵)

Syntax hints:   ∖ cdif 3940   ⊆ wss 3943   × cxp 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5291  ax-nul 5298  ax-pr 5419

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3474  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5141  df-opab 5203  df-xp 5674  df-rel 5675  df-cnv 5676

This theorem is referenced by: (None)