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Theorem conss2 42815
Description: Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
Assertion
Ref Expression
conss2 (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴))

Proof of Theorem conss2
StepHypRef Expression
1 ssv 3972 . 2 𝐴 ⊆ V
2 ssv 3972 . 2 𝐵 ⊆ V
3 ssconb 4101 . 2 ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴)))
41, 2, 3mp2an 691 1 (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  Vcvv 3447  cdif 3911  wss 3914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-dif 3917  df-in 3921  df-ss 3931
This theorem is referenced by: (None)
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