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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 205 Vcvv 3471 ∖ cdif 3944 ⊆ wss 3947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 |
This theorem is referenced by: (None) |
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