Theorem conss2 44414

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 4033	. 2 ⊢ 𝐴 ⊆ V
2		ssv 4033	. 2 ⊢ 𝐵 ⊆ V
3		ssconb 4165	. 2 ⊢ ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴)))
4	1, 2, 3	mp2an 691	1 ⊢ (𝐴 ⊆ (V ∖ 𝐵) ↔ 𝐵 ⊆ (V ∖ 𝐴))

Syntax hints:   ↔ wb 206  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-ss 3993

This theorem is referenced by: (None)