Theorem conss2 44792

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

Step	Hyp	Ref	Expression
1		ssv 3960	. 2 ⊢ 𝐴 ⊆ V
2		ssv 3960	. 2 ⊢ 𝐵 ⊆ V
3		ssconb 4096	. 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴)))
4	1, 2, 3	mp2an 693	1 ⊢ (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴))

Syntax hints:   ↔ wb 206  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-dif 3906  df-ss 3920

This theorem is referenced by: (None)