Theorem ssprr 3839

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 3533	. . . 4 |- (/) C_ { B , C }
2		sseq1 3250	. . . 4 |- ( A = (/) -> ( A C_ { B , C } <-> (/) C_ { B , C } ) )
3	1, 2	mpbiri 168	. . 3 |- ( A = (/) -> A C_ { B , C } )
4		snsspr1 3821	. . . 4 |- { B } C_ { B , C }
5		sseq1 3250	. . . 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 723	. 2 |- ( ( A = (/) \/ A = { B } ) -> A C_ { B , C } )
8		snsspr2 3822	. . . 4 |- { C } C_ { B , C }
9		sseq1 3250	. . . 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 3281	. . 3 |- ( A = { B , C } -> A C_ { B , C } )
12	10, 11	jaoi 723	. 2 |- ( ( A = { C } \/ A = { B , C } ) -> A C_ { B , C } )
13	7, 12	jaoi 723	1 |- ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) -> A C_ { B , C } )

Syntax hints:   -> wi 4   \/ wo 715   = wceq 1397   C_ wss 3200   (/)c0 3494   {csn 3669   {cpr 3670

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pr 3676

This theorem is referenced by:  sstpr  3840  pwprss  3889