Theorem ssdifsn 3761

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 3301	. . 3 |- ( A C_ ( B \ { C } ) -> A C_ B )
2		reldisj 3512	. . . 4 |- ( A C_ B -> ( ( A i^i { C } ) = (/) <-> A C_ ( B \ { C } ) ) )
3	2	bicomd 141	. . 3 |- ( A C_ B -> ( A C_ ( B \ { C } ) <-> ( A i^i { C } ) = (/) ) )
4	1, 3	biadan2 456	. 2 |- ( A C_ ( B \ { C } ) <-> ( A C_ B /\ ( A i^i { C } ) = (/) ) )
5		disjsn 3695	. . 3 |- ( ( A i^i { C } ) = (/) <-> -. C e. A )
6	5	anbi2i 457	. 2 |- ( ( A C_ B /\ ( A i^i { C } ) = (/) ) <-> ( A C_ B /\ -. C e. A ) )
7	4, 6	bitri 184	1 |- ( A C_ ( B \ { C } ) <-> ( A C_ B /\ -. C e. A ) )

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   \ cdif 3163   i^i cin 3165   C_ wss 3166   (/)c0 3460   {csn 3633

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639

This theorem is referenced by: (None)