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Theorem conss2 43716
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 3999 . 2 𝐴 ⊆ V
2 ssv 3999 . 2 𝐵 ⊆ V
3 ssconb 4130 . 2 ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴)))
41, 2, 3mp2an 689 1 (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  Vcvv 3466  cdif 3938  wss 3941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-dif 3944  df-in 3948  df-ss 3958
This theorem is referenced by: (None)
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