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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 3123 | . 2 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐴 ∖ 𝐶)) |
3 | ssdifd.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
4 | 3 | ssdifd 3122 | . 2 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ⊆ (𝐵 ∖ 𝐶)) |
5 | 2, 4 | sstrd 3022 | 1 ⊢ (𝜑 → (𝐴 ∖ 𝐷) ⊆ (𝐵 ∖ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∖ cdif 2983 ⊆ wss 2986 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 |
This theorem depends on definitions: df-bi 115 df-tru 1290 df-nf 1393 df-sb 1690 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-v 2616 df-dif 2988 df-in 2992 df-ss 2999 |
This theorem is referenced by: (None) |
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