Theorem sstpr 3877

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 3876	. . 3 |- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C } )
2		prsstp12 3863	. . 3 |- { B , C } C_ { B , C , D }
3	1, 2	sstrdi 3260	. 2 |- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C , D } )
4		snsstp3 3862	. . . . 5 |- { D } C_ { B , C , D }
5		sseq1 3271	. . . . 5 |- ( A = { D } -> ( A C_ { B , C , D } <-> { D } C_ { B , C , D } ) )
6	4, 5	mpbiri 168	. . . 4 |- ( A = { D } -> A C_ { B , C , D } )
7		prsstp13 3864	. . . . 5 |- { B , D } C_ { B , C , D }
8		sseq1 3271	. . . . 5 |- ( A = { B , D } -> ( A C_ { B , C , D } <-> { B , D } C_ { B , C , D } ) )
9	7, 8	mpbiri 168	. . . 4 |- ( A = { B , D } -> A C_ { B , C , D } )
10	6, 9	jaoi 728	. . 3 |- ( ( A = { D } \/ A = { B , D } ) -> A C_ { B , C , D } )
11		prsstp23 3865	. . . . 5 |- { C , D } C_ { B , C , D }
12		sseq1 3271	. . . . 5 |- ( A = { C , D } -> ( A C_ { B , C , D } <-> { C , D } C_ { B , C , D } ) )
13	11, 12	mpbiri 168	. . . 4 |- ( A = { C , D } -> A C_ { B , C , D } )
14		eqimss 3302	. . . 4 |- ( A = { B , C , D } -> A C_ { B , C , D } )
15	13, 14	jaoi 728	. . 3 |- ( ( A = { C , D } \/ A = { B , C , D } ) -> A C_ { B , C , D } )
16	10, 15	jaoi 728	. 2 |- ( ( ( A = { D } \/ A = { B , D } ) \/ ( A = { C , D } \/ A = { B , C , D } ) ) -> A C_ { B , C , D } )
17	3, 16	jaoi 728	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 720   = wceq 1402   C_ wss 3220   (/)c0 3520   {csn 3705   {cpr 3706   {ctp 3707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3711  df-pr 3712  df-tp 3713

This theorem is referenced by:  pwtpss  3927