Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > difdif2ss | GIF version |
Description: Set difference with a set difference. In classical logic this would be equality rather than subset. (Contributed by Jim Kingdon, 27-Jul-2018.) |
Ref | Expression |
---|---|
difdif2ss | ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∖ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inssdif 3282 | . . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶)) | |
2 | unss2 3217 | . . . 4 ⊢ ((𝐴 ∩ 𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶)) → ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶)))) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) |
4 | difindiss 3300 | . . 3 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) | |
5 | 3, 4 | sstri 3076 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) |
6 | invdif 3288 | . . . 4 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶) | |
7 | 6 | eqcomi 2121 | . . 3 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∩ (V ∖ 𝐶)) |
8 | 7 | difeq2i 3161 | . 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) |
9 | 5, 8 | sseqtrri 3102 | 1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∖ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: Vcvv 2660 ∖ cdif 3038 ∪ cun 3039 ∩ cin 3040 ⊆ wss 3041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |