Theorem ssdif0im 3533

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 3183	. . . 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 1479	. 2 |- ( A. x ( x e. A -> x e. B ) -> A. x -. x e. ( A \ B ) )
5		ssalel 3189	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
6		eq0 3487	. 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 2178   \ cdif 3171   C_ wss 3174   (/)c0 3468

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-nul 3469

This theorem is referenced by:  vdif0im  3534  difrab0eqim  3535  difid  3537  difin0  3542