Theorem ssdifsn 3563

Description: Subset of a set with an element removed. (Contributed by Emmett Weisz, 7-Jul-2021.) (Proof shortened by JJ, 31-May-2022.)

Assertion

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

Proof of Theorem ssdifsn

Step	Hyp	Ref	Expression
1		difss2 3126	. . 3 |- ( A C_ ( B \ { C } ) -> A C_ B )
2		reldisj 3331	. . . 4 |- ( A C_ B -> ( ( A i^i { C } ) = (/) <-> A C_ ( B \ { C } ) ) )
3	2	bicomd 139	. . 3 |- ( A C_ B -> ( A C_ ( B \ { C } ) <-> ( A i^i { C } ) = (/) ) )
4	1, 3	biadan2 444	. 2 |- ( A C_ ( B \ { C } ) <-> ( A C_ B /\ ( A i^i { C } ) = (/) ) )
5		disjsn 3499	. . 3 |- ( ( A i^i { C } ) = (/) <-> -. C e. A )
6	5	anbi2i 445	. 2 |- ( ( A C_ B /\ ( A i^i { C } ) = (/) ) <-> ( A C_ B /\ -. C e. A ) )
7	4, 6	bitri 182	1 |- ( A C_ ( B \ { C } ) <-> ( A C_ B /\ -. C e. A ) )

Syntax hints:   -. wn 3   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   \ cdif 2994   i^i cin 2996   C_ wss 2997   (/)c0 3284   {csn 3441

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-dif 2999  df-in 3003  df-ss 3010  df-nul 3285  df-sn 3447

This theorem is referenced by: (None)