| Mathbox for Andrew Salmon |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > conss1 | Structured version Visualization version GIF version | ||
| Description: Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.) |
| Ref | Expression |
|---|---|
| conss1 | ⊢ ((V ∖ 𝐴) ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | difcom 4489 | 1 ⊢ ((V ∖ 𝐴) ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 |
| This theorem is referenced by: (None) |
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