Theorem sstpr 3684

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 3683	. . 3 |- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C } )
2		prsstp12 3673	. . 3 |- { B , C } C_ { B , C , D }
3	1, 2	sstrdi 3109	. 2 |- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C , D } )
4		snsstp3 3672	. . . . 5 |- { D } C_ { B , C , D }
5		sseq1 3120	. . . . 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 3674	. . . . 5 |- { B , D } C_ { B , C , D }
8		sseq1 3120	. . . . 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 705	. . 3 |- ( ( A = { D } \/ A = { B , D } ) -> A C_ { B , C , D } )
11		prsstp23 3675	. . . . 5 |- { C , D } C_ { B , C , D }
12		sseq1 3120	. . . . 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 3151	. . . 4 |- ( A = { B , C , D } -> A C_ { B , C , D } )
15	13, 14	jaoi 705	. . 3 |- ( ( A = { C , D } \/ A = { B , C , D } ) -> A C_ { B , C , D } )
16	10, 15	jaoi 705	. 2 |- ( ( ( A = { D } \/ A = { B , D } ) \/ ( A = { C , D } \/ A = { B , C , D } ) ) -> A C_ { B , C , D } )
17	3, 16	jaoi 705	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 697   = wceq 1331   C_ wss 3071   (/)c0 3363   {csn 3527   {cpr 3528   {ctp 3529

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3or 963  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-tp 3535

This theorem is referenced by:  pwtpss  3733