Theorem ssdifsn 3659

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 3209	. . 3 |- ( A C_ ( B \ { C } ) -> A C_ B )
2		reldisj 3419	. . . 4 |- ( A C_ B -> ( ( A i^i { C } ) = (/) <-> A C_ ( B \ { C } ) ) )
3	2	bicomd 140	. . 3 |- ( A C_ B -> ( A C_ ( B \ { C } ) <-> ( A i^i { C } ) = (/) ) )
4	1, 3	biadan2 452	. 2 |- ( A C_ ( B \ { C } ) <-> ( A C_ B /\ ( A i^i { C } ) = (/) ) )
5		disjsn 3593	. . 3 |- ( ( A i^i { C } ) = (/) <-> -. C e. A )
6	5	anbi2i 453	. 2 |- ( ( A C_ B /\ ( A i^i { C } ) = (/) ) <-> ( A C_ B /\ -. C e. A ) )
7	4, 6	bitri 183	1 |- ( A C_ ( B \ { C } ) <-> ( A C_ B /\ -. C e. A ) )

Syntax hints:   -. wn 3   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   \ cdif 3073   i^i cin 3075   C_ wss 3076   (/)c0 3368   {csn 3532

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-nul 3369  df-sn 3538

This theorem is referenced by: (None)