Theorem ssdifsn 4318

Description: Subset of a set with an element removed. (Contributed by Emmett Weisz, 7-Jul-2021.)

Assertion

Ref	Expression
ssdifsn	|- ( A C_ ( B \ { C } ) <-> ( A C_ B /\ -. C e. A ) )

Proof of Theorem ssdifsn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss3 3592	. . . 4 |- ( A C_ ( B \ { C } ) <-> A. x e. A x e. ( B \ { C } ) )
2		eldifsn 4317	. . . . 5 |- ( x e. ( B \ { C } ) <-> ( x e. B /\ x =/= C ) )
3	2	ralbii 2980	. . . 4 |- ( A. x e. A x e. ( B \ { C } ) <-> A. x e. A ( x e. B /\ x =/= C ) )
4	1, 3	bitri 264	. . 3 |- ( A C_ ( B \ { C } ) <-> A. x e. A ( x e. B /\ x =/= C ) )
5		r19.26 3064	. . 3 |- ( A. x e. A ( x e. B /\ x =/= C ) <-> ( A. x e. A x e. B /\ A. x e. A x =/= C ) )
6	4, 5	bitri 264	. 2 |- ( A C_ ( B \ { C } ) <-> ( A. x e. A x e. B /\ A. x e. A x =/= C ) )
7		dfss3 3592	. . . 4 |- ( A C_ B <-> A. x e. A x e. B )
8	7	bicomi 214	. . 3 |- ( A. x e. A x e. B <-> A C_ B )
9		neirr 2803	. . . . 5 |- -. C =/= C
10		neeq1 2856	. . . . . 6 |- ( x = C -> ( x =/= C <-> C =/= C ) )
11	10	rspccv 3306	. . . . 5 |- ( A. x e. A x =/= C -> ( C e. A -> C =/= C ) )
12	9, 11	mtoi 190	. . . 4 |- ( A. x e. A x =/= C -> -. C e. A )
13		nelelne 2892	. . . . 5 |- ( -. C e. A -> ( x e. A -> x =/= C ) )
14	13	ralrimiv 2965	. . . 4 |- ( -. C e. A -> A. x e. A x =/= C )
15	12, 14	impbii 199	. . 3 |- ( A. x e. A x =/= C <-> -. C e. A )
16	8, 15	anbi12i 733	. 2 |- ( ( A. x e. A x e. B /\ A. x e. A x =/= C ) <-> ( A C_ B /\ -. C e. A ) )
17	6, 16	bitri 264	1 |- ( A C_ ( B \ { C } ) <-> ( A C_ B /\ -. C e. A ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   C_ wss 3574   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-sn 4178

This theorem is referenced by:  logdivsqrle  30728  elsetrecslem  42444