Theorem ssdifsn 3801

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 3335	. . 3 |- ( A C_ ( B \ { C } ) -> A C_ B )
2		reldisj 3546	. . . 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 3731	. . 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 1397   e. wcel 2202   \ cdif 3197   i^i cin 3199   C_ wss 3200   (/)c0 3494   {csn 3669

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675

This theorem is referenced by: (None)