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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 3307 | . . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶)) | |
2 | unss2 3242 | . . . 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 3325 | . . 3 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) | |
5 | 3, 4 | sstri 3101 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) |
6 | invdif 3313 | . . . 4 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶) | |
7 | 6 | eqcomi 2141 | . . 3 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∩ (V ∖ 𝐶)) |
8 | 7 | difeq2i 3186 | . 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) |
9 | 5, 8 | sseqtrri 3127 | 1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∖ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: Vcvv 2681 ∖ cdif 3063 ∪ cun 3064 ∩ cin 3065 ⊆ wss 3066 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 |
This theorem is referenced by: (None) |
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