Theorem difxp1ss 32905

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 6167	. 2 ⊢ ((𝐴 ∖ 𝐶) × 𝐵) = ((𝐴 × 𝐵) ∖ (𝐶 × 𝐵))
2		difss 4093	. 2 ⊢ ((𝐴 × 𝐵) ∖ (𝐶 × 𝐵)) ⊆ (𝐴 × 𝐵)
3	1, 2	eqsstri 3986	1 ⊢ ((𝐴 ∖ 𝐶) × 𝐵) ⊆ (𝐴 × 𝐵)

This proof depends on syntax axioms:   ∖ cdif 3905   ⊆ wss 3908   × cxp 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-opab 5179  df-xp 5672  df-rel 5673

This theorem is used by: (None)