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Mirrors > Home > ILE Home > Th. List > ssdifss | GIF version |
Description: Preservation of a subclass relationship by class difference. (Contributed by NM, 15-Feb-2007.) |
Ref | Expression |
---|---|
ssdifss | ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 3172 | . 2 ⊢ (𝐴 ∖ 𝐶) ⊆ 𝐴 | |
2 | sstr 3075 | . 2 ⊢ (((𝐴 ∖ 𝐶) ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ∖ 𝐶) ⊆ 𝐵) | |
3 | 1, 2 | mpan 420 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∖ 𝐶) ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∖ cdif 3038 ⊆ 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-v 2662 df-dif 3043 df-in 3047 df-ss 3054 |
This theorem is referenced by: ssdifssd 3184 |
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