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Theorem recnprss 12834
Description: Both  RR and  CC are subsets of  CC. (Contributed by Mario Carneiro, 10-Feb-2015.)
Assertion
Ref Expression
recnprss  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )

Proof of Theorem recnprss
StepHypRef Expression
1 elpri 3550 . 2  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
2 ax-resscn 7719 . . . 4  |-  RR  C_  CC
3 sseq1 3120 . . . 4  |-  ( S  =  RR  ->  ( S  C_  CC  <->  RR  C_  CC ) )
42, 3mpbiri 167 . . 3  |-  ( S  =  RR  ->  S  C_  CC )
5 eqimss 3151 . . 3  |-  ( S  =  CC  ->  S  C_  CC )
64, 5jaoi 705 . 2  |-  ( ( S  =  RR  \/  S  =  CC )  ->  S  C_  CC )
71, 6syl 14 1  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 697    = wceq 1331    e. wcel 1480    C_ wss 3071   {cpr 3528   CCcc 7625   RRcr 7626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-resscn 7719
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534
This theorem is referenced by:  dvfgg  12835  dvaddxx  12845  dvmulxx  12846  dviaddf  12847  dvimulf  12848
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