Theorem sstpr 3736

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

Assertion

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

Proof of Theorem sstpr

Step	Hyp	Ref	Expression
1		ssprr 3735	. . 3 |- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C } )
2		prsstp12 3725	. . 3 |- { B , C } C_ { B , C , D }
3	1, 2	sstrdi 3153	. 2 |- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C , D } )
4		snsstp3 3724	. . . . 5 |- { D } C_ { B , C , D }
5		sseq1 3164	. . . . 5 |- ( A = { D } -> ( A C_ { B , C , D } <-> { D } C_ { B , C , D } ) )
6	4, 5	mpbiri 167	. . . 4 |- ( A = { D } -> A C_ { B , C , D } )
7		prsstp13 3726	. . . . 5 |- { B , D } C_ { B , C , D }
8		sseq1 3164	. . . . 5 |- ( A = { B , D } -> ( A C_ { B , C , D } <-> { B , D } C_ { B , C , D } ) )
9	7, 8	mpbiri 167	. . . 4 |- ( A = { B , D } -> A C_ { B , C , D } )
10	6, 9	jaoi 706	. . 3 |- ( ( A = { D } \/ A = { B , D } ) -> A C_ { B , C , D } )
11		prsstp23 3727	. . . . 5 |- { C , D } C_ { B , C , D }
12		sseq1 3164	. . . . 5 |- ( A = { C , D } -> ( A C_ { B , C , D } <-> { C , D } C_ { B , C , D } ) )
13	11, 12	mpbiri 167	. . . 4 |- ( A = { C , D } -> A C_ { B , C , D } )
14		eqimss 3195	. . . 4 |- ( A = { B , C , D } -> A C_ { B , C , D } )
15	13, 14	jaoi 706	. . 3 |- ( ( A = { C , D } \/ A = { B , C , D } ) -> A C_ { B , C , D } )
16	10, 15	jaoi 706	. 2 |- ( ( ( A = { D } \/ A = { B , D } ) \/ ( A = { C , D } \/ A = { B , C , D } ) ) -> A C_ { B , C , D } )
17	3, 16	jaoi 706	1 |- ( ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) \/ ( ( A = { D } \/ A = { B , D } ) \/ ( A = { C , D } \/ A = { B , C , D } ) ) ) -> A C_ { B , C , D } )

Syntax hints:   -> wi 4   \/ wo 698   = wceq 1343   C_ wss 3115   (/)c0 3408   {csn 3575   {cpr 3576   {ctp 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-3or 969  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-sn 3581  df-pr 3582  df-tp 3583

This theorem is referenced by:  pwtpss  3785