Theorem ssprr 3805

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

Assertion

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

Proof of Theorem ssprr

Step	Hyp	Ref	Expression
1		0ss 3503	. . . 4 |- (/) C_ { B , C }
2		sseq1 3220	. . . 4 |- ( A = (/) -> ( A C_ { B , C } <-> (/) C_ { B , C } ) )
3	1, 2	mpbiri 168	. . 3 |- ( A = (/) -> A C_ { B , C } )
4		snsspr1 3787	. . . 4 |- { B } C_ { B , C }
5		sseq1 3220	. . . 4 |- ( A = { B } -> ( A C_ { B , C } <-> { B } C_ { B , C } ) )
6	4, 5	mpbiri 168	. . 3 |- ( A = { B } -> A C_ { B , C } )
7	3, 6	jaoi 718	. 2 |- ( ( A = (/) \/ A = { B } ) -> A C_ { B , C } )
8		snsspr2 3788	. . . 4 |- { C } C_ { B , C }
9		sseq1 3220	. . . 4 |- ( A = { C } -> ( A C_ { B , C } <-> { C } C_ { B , C } ) )
10	8, 9	mpbiri 168	. . 3 |- ( A = { C } -> A C_ { B , C } )
11		eqimss 3251	. . 3 |- ( A = { B , C } -> A C_ { B , C } )
12	10, 11	jaoi 718	. 2 |- ( ( A = { C } \/ A = { B , C } ) -> A C_ { B , C } )
13	7, 12	jaoi 718	1 |- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C } )

Syntax hints:   -> wi 4   \/ wo 710   = wceq 1373   C_ wss 3170   (/)c0 3464   {csn 3638   {cpr 3639

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pr 3645

This theorem is referenced by:  sstpr  3806  pwprss  3855