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Mirrors > Home > MPE Home > Th. List > difxp1 | Structured version Visualization version GIF version |
Description: Difference law for Cartesian product. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 26-Jun-2014.) |
Ref | Expression |
---|---|
difxp1 | ⊢ ((𝐴 ∖ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∖ (𝐵 × 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difxp 6021 | . 2 ⊢ ((𝐴 × 𝐶) ∖ (𝐵 × 𝐶)) = (((𝐴 ∖ 𝐵) × 𝐶) ∪ (𝐴 × (𝐶 ∖ 𝐶))) | |
2 | difid 4330 | . . . . 5 ⊢ (𝐶 ∖ 𝐶) = ∅ | |
3 | 2 | xpeq2i 5582 | . . . 4 ⊢ (𝐴 × (𝐶 ∖ 𝐶)) = (𝐴 × ∅) |
4 | xp0 6015 | . . . 4 ⊢ (𝐴 × ∅) = ∅ | |
5 | 3, 4 | eqtri 2844 | . . 3 ⊢ (𝐴 × (𝐶 ∖ 𝐶)) = ∅ |
6 | 5 | uneq2i 4136 | . 2 ⊢ (((𝐴 ∖ 𝐵) × 𝐶) ∪ (𝐴 × (𝐶 ∖ 𝐶))) = (((𝐴 ∖ 𝐵) × 𝐶) ∪ ∅) |
7 | un0 4344 | . 2 ⊢ (((𝐴 ∖ 𝐵) × 𝐶) ∪ ∅) = ((𝐴 ∖ 𝐵) × 𝐶) | |
8 | 1, 6, 7 | 3eqtrri 2849 | 1 ⊢ ((𝐴 ∖ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∖ (𝐵 × 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3933 ∪ cun 3934 ∅c0 4291 × cxp 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 |
This theorem is referenced by: difxp1ss 30282 sxbrsigalem2 31544 |
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