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Mirrors > Home > ILE Home > Th. List > ssindif0im | GIF version |
Description: Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.) |
Ref | Expression |
---|---|
ssindif0im | ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (V ∖ 𝐵)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ddifss 3309 | . . 3 ⊢ 𝐵 ⊆ (V ∖ (V ∖ 𝐵)) | |
2 | sstr 3100 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ (V ∖ (V ∖ 𝐵))) → 𝐴 ⊆ (V ∖ (V ∖ 𝐵))) | |
3 | 1, 2 | mpan2 421 | . 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (V ∖ (V ∖ 𝐵))) |
4 | disj2 3413 | . 2 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵))) | |
5 | 3, 4 | sylibr 133 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (V ∖ 𝐵)) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 Vcvv 2681 ∖ cdif 3063 ∩ cin 3065 ⊆ wss 3066 ∅c0 3358 |
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-v 2683 df-dif 3068 df-in 3072 df-ss 3079 df-nul 3359 |
This theorem is referenced by: (None) |
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