Theorem ssdif0im 3479

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 683	. . . 4 |- ( ( x e. A -> x e. B ) -> -. ( x e. A /\ -. x e. B ) )
2		eldif 3130	. . . 4 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
3	1, 2	sylnibr 672	. . 3 |- ( ( x e. A -> x e. B ) -> -. x e. ( A \ B ) )
4	3	alimi 1448	. 2 |- ( A. x ( x e. A -> x e. B ) -> A. x -. x e. ( A \ B ) )
5		dfss2 3136	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
6		eq0 3433	. 2 |- ( ( A \ B ) = (/) <-> A. x -. x e. ( A \ B ) )
7	4, 5, 6	3imtr4i 200	1 |- ( A C_ B -> ( A \ B ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1346   = wceq 1348   e. wcel 2141   \ cdif 3118   C_ wss 3121   (/)c0 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415

This theorem is referenced by:  vdif0im  3480  difrab0eqim  3481  difid  3483  difin0  3488