Theorem ssdif0im 3561

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 695	. . . 4 |- ( ( x e. A -> x e. B ) -> -. ( x e. A /\ -. x e. B ) )
2		eldif 3210	. . . 4 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
3	1, 2	sylnibr 684	. . 3 |- ( ( x e. A -> x e. B ) -> -. x e. ( A \ B ) )
4	3	alimi 1504	. 2 |- ( A. x ( x e. A -> x e. B ) -> A. x -. x e. ( A \ B ) )
5		ssalel 3216	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
6		eq0 3515	. 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 1396   = wceq 1398   e. wcel 2202   \ cdif 3198   C_ wss 3201   (/)c0 3496

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-nul 3497

This theorem is referenced by:  vdif0im  3562  difrab0eqim  3563  difid  3565  difin0  3570