Theorem ssdif0im 3525

Description: Subclass implies empty difference. One direction of Exercise 7 of [TakeutiZaring] p. 22. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 2-Aug-2018.)

Assertion

Ref	Expression
ssdif0im	|- ( A C_ B -> ( A \ B ) = (/) )

Proof of Theorem ssdif0im

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imanim 690	. . . 4 |- ( ( x e. A -> x e. B ) -> -. ( x e. A /\ -. x e. B ) )
2		eldif 3175	. . . 4 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
3	1, 2	sylnibr 679	. . 3 |- ( ( x e. A -> x e. B ) -> -. x e. ( A \ B ) )
4	3	alimi 1478	. 2 |- ( A. x ( x e. A -> x e. B ) -> A. x -. x e. ( A \ B ) )
5		ssalel 3181	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
6		eq0 3479	. 2 |- ( ( A \ B ) = (/) <-> A. x -. x e. ( A \ B ) )
7	4, 5, 6	3imtr4i 201	1 |- ( A C_ B -> ( A \ B ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1371   = wceq 1373   e. wcel 2176   \ cdif 3163   C_ wss 3166   (/)c0 3460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3461

This theorem is referenced by:  vdif0im  3526  difrab0eqim  3527  difid  3529  difin0  3534