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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 4033 | . 2 ⊢ 𝐴 ⊆ V | |
2 | ssv 4033 | . 2 ⊢ 𝐵 ⊆ V | |
3 | ssconb 4165 | . 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴))) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 Vcvv 3488 ∖ cdif 3973 ⊆ wss 3976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-v 3490 df-dif 3979 df-ss 3993 |
This theorem is referenced by: (None) |
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