Theorem ssprr 3876

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 3561	. . . 4 |- (/) C_ { B , C }
2		sseq1 3271	. . . 4 |- ( A = (/) -> ( A C_ { B , C } <-> (/) C_ { B , C } ) )
3	1, 2	mpbiri 168	. . 3 |- ( A = (/) -> A C_ { B , C } )
4		snsspr1 3858	. . . 4 |- { B } C_ { B , C }
5		sseq1 3271	. . . 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 728	. 2 |- ( ( A = (/) \/ A = { B } ) -> A C_ { B , C } )
8		snsspr2 3859	. . . 4 |- { C } C_ { B , C }
9		sseq1 3271	. . . 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 3302	. . 3 |- ( A = { B , C } -> A C_ { B , C } )
12	10, 11	jaoi 728	. 2 |- ( ( A = { C } \/ A = { B , C } ) -> A C_ { B , C } )
13	7, 12	jaoi 728	1 |- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C } )

Syntax hints:   -> wi 4   \/ wo 720   = wceq 1402   C_ wss 3220   (/)c0 3520   {csn 3705   {cpr 3706

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-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-pr 3712

This theorem is referenced by:  sstpr  3877  pwprss  3926