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Theorem ssrecnpr 29743
Description:  RR is a subset of both  RR and  CC. (Contributed by Steve Rodriguez, 22-Nov-2015.)
Assertion
Ref Expression
ssrecnpr  |-  ( S  e.  { RR ,  CC }  ->  RR  C_  S
)

Proof of Theorem ssrecnpr
StepHypRef Expression
1 elpri 4006 . 2  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
2 eqimss2 3518 . . 3  |-  ( S  =  RR  ->  RR  C_  S )
3 ax-resscn 9451 . . . 4  |-  RR  C_  CC
4 sseq2 3487 . . . 4  |-  ( S  =  CC  ->  ( RR  C_  S  <->  RR  C_  CC ) )
53, 4mpbiri 233 . . 3  |-  ( S  =  CC  ->  RR  C_  S )
62, 5jaoi 379 . 2  |-  ( ( S  =  RR  \/  S  =  CC )  ->  RR  C_  S )
71, 6syl 16 1  |-  ( S  e.  { RR ,  CC }  ->  RR  C_  S
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1370    e. wcel 1758    C_ wss 3437   {cpr 3988   CCcc 9392   RRcr 9393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-resscn 9451
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-un 3442  df-in 3444  df-ss 3451  df-sn 3987  df-pr 3989
This theorem is referenced by: (None)
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