Theorem conss2 45628

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 3937	. 2 ⊢ 𝐴 ⊆ V
2		ssv 3937	. 2 ⊢ 𝐵 ⊆ V
3		ssconb 4081	. 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴)))
4	1, 2, 3	mp2an 705	1 ⊢ (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴))

This proof depends on syntax axioms:   ↔ wb 209  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-dif 4050

This theorem is used by: (None)