Theorem sstpr 3549

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 3548	. . 3 |- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C } )
2		prsstp12 3538	. . 3 |- { B , C } C_ { B , C , D }
3	1, 2	syl6ss 3011	. 2 |- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C , D } )
4		snsstp3 3537	. . . . 5 |- { D } C_ { B , C , D }
5		sseq1 3020	. . . . 5 |- ( A = { D } -> ( A C_ { B , C , D } <-> { D } C_ { B , C , D } ) )
6	4, 5	mpbiri 166	. . . 4 |- ( A = { D } -> A C_ { B , C , D } )
7		prsstp13 3539	. . . . 5 |- { B , D } C_ { B , C , D }
8		sseq1 3020	. . . . 5 |- ( A = { B , D } -> ( A C_ { B , C , D } <-> { B , D } C_ { B , C , D } ) )
9	7, 8	mpbiri 166	. . . 4 |- ( A = { B , D } -> A C_ { B , C , D } )
10	6, 9	jaoi 668	. . 3 |- ( ( A = { D } \/ A = { B , D } ) -> A C_ { B , C , D } )
11		prsstp23 3540	. . . . 5 |- { C , D } C_ { B , C , D }
12		sseq1 3020	. . . . 5 |- ( A = { C , D } -> ( A C_ { B , C , D } <-> { C , D } C_ { B , C , D } ) )
13	11, 12	mpbiri 166	. . . 4 |- ( A = { C , D } -> A C_ { B , C , D } )
14		eqimss 3051	. . . 4 |- ( A = { B , C , D } -> A C_ { B , C , D } )
15	13, 14	jaoi 668	. . 3 |- ( ( A = { C , D } \/ A = { B , C , D } ) -> A C_ { B , C , D } )
16	10, 15	jaoi 668	. 2 |- ( ( ( A = { D } \/ A = { B , D } ) \/ ( A = { C , D } \/ A = { B , C , D } ) ) -> A C_ { B , C , D } )
17	3, 16	jaoi 668	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 661   = wceq 1284   C_ wss 2973   (/)c0 3251   {csn 3398   {cpr 3399   {ctp 3400

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3or 920  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-sn 3404  df-pr 3405  df-tp 3406

This theorem is referenced by:  pwtpss  3598