Mathbox for Andrew Salmon |
< Previous
Next >
Nearby theorems |
||
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 3625 | . 2 ⊢ 𝐴 ⊆ V | |
2 | ssv 3625 | . 2 ⊢ 𝐵 ⊆ V | |
3 | ssconb 3743 | . 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴))) | |
4 | 1, 2, 3 | mp2an 708 | 1 ⊢ (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |