![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > complss | Structured version Visualization version GIF version |
Description: Complementation reverses inclusion. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 19-Mar-2021.) |
Ref | Expression |
---|---|
complss | ⊢ (𝐴 ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sscon 3966 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (V ∖ 𝐵) ⊆ (V ∖ 𝐴)) | |
2 | sscon 3966 | . . 3 ⊢ ((V ∖ 𝐵) ⊆ (V ∖ 𝐴) → (V ∖ (V ∖ 𝐴)) ⊆ (V ∖ (V ∖ 𝐵))) | |
3 | ddif 3964 | . . 3 ⊢ (V ∖ (V ∖ 𝐴)) = 𝐴 | |
4 | ddif 3964 | . . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵 | |
5 | 2, 3, 4 | 3sstr3g 3863 | . 2 ⊢ ((V ∖ 𝐵) ⊆ (V ∖ 𝐴) → 𝐴 ⊆ 𝐵) |
6 | 1, 5 | impbii 201 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 Vcvv 3397 ∖ cdif 3788 ⊆ wss 3791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-v 3399 df-dif 3794 df-in 3798 df-ss 3805 |
This theorem is referenced by: compleq 3974 |
Copyright terms: Public domain | W3C validator |