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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 4131 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (V ∖ 𝐵) ⊆ (V ∖ 𝐴)) | |
2 | sscon 4131 | . . 3 ⊢ ((V ∖ 𝐵) ⊆ (V ∖ 𝐴) → (V ∖ (V ∖ 𝐴)) ⊆ (V ∖ (V ∖ 𝐵))) | |
3 | ddif 4129 | . . 3 ⊢ (V ∖ (V ∖ 𝐴)) = 𝐴 | |
4 | ddif 4129 | . . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵 | |
5 | 2, 3, 4 | 3sstr3g 4019 | . 2 ⊢ ((V ∖ 𝐵) ⊆ (V ∖ 𝐴) → 𝐴 ⊆ 𝐵) |
6 | 1, 5 | impbii 208 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 Vcvv 3466 ∖ cdif 3938 ⊆ wss 3941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-dif 3944 df-in 3948 df-ss 3958 |
This theorem is referenced by: compleq 4140 |
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