Theorem ssprr 3736

Description: The subsets of a pair. (Contributed by Jim Kingdon, 11-Aug-2018.)

Assertion

Ref	Expression
ssprr	|- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C } )

Proof of Theorem ssprr

Step	Hyp	Ref	Expression
1		0ss 3447	. . . 4 |- (/) C_ { B , C }
2		sseq1 3165	. . . 4 |- ( A = (/) -> ( A C_ { B , C } <-> (/) C_ { B , C } ) )
3	1, 2	mpbiri 167	. . 3 |- ( A = (/) -> A C_ { B , C } )
4		snsspr1 3721	. . . 4 |- { B } C_ { B , C }
5		sseq1 3165	. . . 4 |- ( A = { B } -> ( A C_ { B , C } <-> { B } C_ { B , C } ) )
6	4, 5	mpbiri 167	. . 3 |- ( A = { B } -> A C_ { B , C } )
7	3, 6	jaoi 706	. 2 |- ( ( A = (/) \/ A = { B } ) -> A C_ { B , C } )
8		snsspr2 3722	. . . 4 |- { C } C_ { B , C }
9		sseq1 3165	. . . 4 |- ( A = { C } -> ( A C_ { B , C } <-> { C } C_ { B , C } ) )
10	8, 9	mpbiri 167	. . 3 |- ( A = { C } -> A C_ { B , C } )
11		eqimss 3196	. . 3 |- ( A = { B , C } -> A C_ { B , C } )
12	10, 11	jaoi 706	. 2 |- ( ( A = { C } \/ A = { B , C } ) -> A C_ { B , C } )
13	7, 12	jaoi 706	1 |- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C } )

Syntax hints:   -> wi 4   \/ wo 698   = wceq 1343   C_ wss 3116   (/)c0 3409   {csn 3576   {cpr 3577

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pr 3583

This theorem is referenced by:  sstpr  3737  pwprss  3785