Theorem conss2 38647

Description: Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)

Assertion

Ref	Expression
conss2	|- ( A C_ ( _V \ B ) <-> B C_ ( _V \ A ) )

Proof of Theorem conss2

Step	Hyp	Ref	Expression
1		ssv 3625	. 2 |- A C_ _V
2		ssv 3625	. 2 |- B C_ _V
3		ssconb 3743	. 2 |- ( ( A C_ _V /\ B C_ _V ) -> ( A C_ ( _V \ B ) <-> B C_ ( _V \ A ) ) )
4	1, 2, 3	mp2an 708	1 |- ( A C_ ( _V \ B ) <-> B C_ ( _V \ A ) )

Syntax hints:   <-> wb 196   _Vcvv 3200   \ cdif 3571   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588

This theorem is referenced by: (None)