Theorem difxp2ss 32458

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 6141	. 2 ⊢ (𝐴 × (𝐵 ∖ 𝐶)) = ((𝐴 × 𝐵) ∖ (𝐴 × 𝐶))
2		difss 4101	. 2 ⊢ ((𝐴 × 𝐵) ∖ (𝐴 × 𝐶)) ⊆ (𝐴 × 𝐵)
3	1, 2	eqsstri 3995	1 ⊢ (𝐴 × (𝐵 ∖ 𝐶)) ⊆ (𝐴 × 𝐵)

Syntax hints:   ∖ cdif 3913   ⊆ wss 3916   × cxp 5638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5172  df-xp 5646  df-rel 5647

This theorem is referenced by: (None)