Theorem ssprr 3691

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 3406	. . . 4 |- (/) C_ { B , C }
2		sseq1 3125	. . . 4 |- ( A = (/) -> ( A C_ { B , C } <-> (/) C_ { B , C } ) )
3	1, 2	mpbiri 167	. . 3 |- ( A = (/) -> A C_ { B , C } )
4		snsspr1 3676	. . . 4 |- { B } C_ { B , C }
5		sseq1 3125	. . . 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 3677	. . . 4 |- { C } C_ { B , C }
9		sseq1 3125	. . . 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 3156	. . 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 1332   C_ wss 3076   (/)c0 3368   {csn 3532   {cpr 3533

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-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pr 3539

This theorem is referenced by:  sstpr  3692  pwprss  3740