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Mirrors > Home > ILE Home > Th. List > ssdif2d | GIF version |
Description: If 𝐴 is contained in 𝐵 and 𝐶 is contained in 𝐷, then (𝐴 ∖ 𝐷) is contained in (𝐵 ∖ 𝐶). Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
ssdifd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
ssdif2d.2 | ⊢ (𝜑 → 𝐶 ⊆ 𝐷) |
Ref | Expression |
---|---|
ssdif2d | ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐵 ∖ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdif2d.2 | . . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐷) | |
2 | 1 | sscond 3272 | . 2 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐴 ∖ 𝐶)) |
3 | ssdifd.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
4 | 3 | ssdifd 3271 | . 2 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) |
5 | 2, 4 | sstrd 3165 | 1 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐵 ∖ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∖ cdif 3126 ⊆ wss 3129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-dif 3131 df-in 3135 df-ss 3142 |
This theorem is referenced by: (None) |
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