![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > difxp2ss | Structured version Visualization version GIF version |
Description: Difference law for Cartesian products. (Contributed by Thierry Arnoux, 24-Jul-2023.) |
Ref | Expression |
---|---|
difxp2ss | ⊢ (𝐴 × (𝐵 ∖ 𝐶)) ⊆ (𝐴 × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difxp2 6172 | . 2 ⊢ (𝐴 × (𝐵 ∖ 𝐶)) = ((𝐴 × 𝐵) ∖ (𝐴 × 𝐶)) | |
2 | difss 4128 | . 2 ⊢ ((𝐴 × 𝐵) ∖ (𝐴 × 𝐶)) ⊆ (𝐴 × 𝐵) | |
3 | 1, 2 | eqsstri 4011 | 1 ⊢ (𝐴 × (𝐵 ∖ 𝐶)) ⊆ (𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∖ cdif 3941 ⊆ wss 3944 × cxp 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-opab 5212 df-xp 5684 df-rel 5685 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |