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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 3588 | . 2 ⊢ 𝐴 ⊆ V | |
2 | ssv 3588 | . 2 ⊢ 𝐵 ⊆ V | |
3 | ssconb 3705 | . 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴))) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 |
This theorem is referenced by: (None) |
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