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Mirrors > Home > MPE Home > Th. List > Mathboxes > conss2 | Structured version Visualization version GIF version |
Description: Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.) |
Ref | Expression |
---|---|
conss2 | ⊢ (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3994 | . 2 ⊢ 𝐴 ⊆ V | |
2 | ssv 3994 | . 2 ⊢ 𝐵 ⊆ V | |
3 | ssconb 4117 | . 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴))) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 Vcvv 3497 ∖ cdif 3936 ⊆ wss 3939 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-dif 3942 df-in 3946 df-ss 3955 |
This theorem is referenced by: (None) |
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