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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 4488 | 1 ⊢ ((V ∖ 𝐴) ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 Vcvv 3474 ∖ cdif 3945 ⊆ wss 3948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 |
This theorem is referenced by: (None) |
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