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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 3354 | . . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ (𝐴 ∖ (V ∖ 𝐶)) | |
2 | unss2 3289 | . . . 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 3372 | . . 3 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) | |
5 | 3, 4 | sstri 3147 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) |
6 | invdif 3360 | . . . 4 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶) | |
7 | 6 | eqcomi 2168 | . . 3 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∩ (V ∖ 𝐶)) |
8 | 7 | difeq2i 3233 | . 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) |
9 | 5, 8 | sseqtrri 3173 | 1 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) ⊆ (𝐴 ∖ (𝐵 ∖ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: Vcvv 2722 ∖ cdif 3109 ∪ cun 3110 ∩ cin 3111 ⊆ wss 3112 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rab 2451 df-v 2724 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 |
This theorem is referenced by: (None) |
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