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Theorem ssrecnpr 27515
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 3835 . 2  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
2 eqimss2 3402 . . 3  |-  ( S  =  RR  ->  RR  C_  S )
3 ax-resscn 9048 . . . 4  |-  RR  C_  CC
4 sseq2 3371 . . . 4  |-  ( S  =  CC  ->  ( RR  C_  S  <->  RR  C_  CC ) )
53, 4mpbiri 226 . . 3  |-  ( S  =  CC  ->  RR  C_  S )
62, 5jaoi 370 . 2  |-  ( ( S  =  RR  \/  S  =  CC )  ->  RR  C_  S )
71, 6syl 16 1  |-  ( S  e.  { RR ,  CC }  ->  RR  C_  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    = wceq 1653    e. wcel 1726    C_ wss 3321   {cpr 3816   CCcc 8989   RRcr 8990
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-resscn 9048
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-v 2959  df-un 3326  df-in 3328  df-ss 3335  df-sn 3821  df-pr 3822
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