Theorem sstpr 3692

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 3691	. . 3 |- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C } )
2		prsstp12 3681	. . 3 |- { B , C } C_ { B , C , D }
3	1, 2	sstrdi 3114	. 2 |- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C , D } )
4		snsstp3 3680	. . . . 5 |- { D } C_ { B , C , D }
5		sseq1 3125	. . . . 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 3682	. . . . 5 |- { B , D } C_ { B , C , D }
8		sseq1 3125	. . . . 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 3683	. . . . 5 |- { C , D } C_ { B , C , D }
12		sseq1 3125	. . . . 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 3156	. . . 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 1332   C_ wss 3076   (/)c0 3368   {csn 3532   {cpr 3533   {ctp 3534

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-3or 964  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-sn 3538  df-pr 3539  df-tp 3540

This theorem is referenced by:  pwtpss  3741