Theorem ssindif0im 3281

Description: Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)

Assertion

Ref	Expression
ssindif0im	|- ( A C_ B -> ( A i^i ( _V \ B ) ) = (/) )

Proof of Theorem ssindif0im

Step	Hyp	Ref	Expression
1		ddifss 3175	. . 3 |- B C_ ( _V \ ( _V \ B ) )
2		sstr 2953	. . 3 |- ( ( A C_ B /\ B C_ ( _V \ ( _V \ B ) ) ) -> A C_ ( _V \ ( _V \ B ) ) )
3	1, 2	mpan2 401	. 2 |- ( A C_ B -> A C_ ( _V \ ( _V \ B ) ) )
4		disj2 3275	. 2 |- ( ( A i^i ( _V \ B ) ) = (/) <-> A C_ ( _V \ ( _V \ B ) ) )
5	3, 4	sylibr 137	1 |- ( A C_ B -> ( A i^i ( _V \ B ) ) = (/) )

Syntax hints:   -> wi 4   = wceq 1243   _Vcvv 2557   \ cdif 2914   i^i cin 2916   C_ wss 2917   (/)c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225

This theorem is referenced by: (None)