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Theorem ssrecnpr 27537
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 3660 . 2  |-  ( S  e.  { RR ,  CC }  ->  ( S  =  RR  \/  S  =  CC ) )
2 eqimss2 3231 . . 3  |-  ( S  =  RR  ->  RR  C_  S )
3 ax-resscn 8794 . . . 4  |-  RR  C_  CC
4 sseq2 3200 . . . 4  |-  ( S  =  CC  ->  ( RR  C_  S  <->  RR  C_  CC ) )
53, 4mpbiri 224 . . 3  |-  ( S  =  CC  ->  RR  C_  S )
62, 5jaoi 368 . 2  |-  ( ( S  =  RR  \/  S  =  CC )  ->  RR  C_  S )
71, 6syl 15 1  |-  ( S  e.  { RR ,  CC }  ->  RR  C_  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    = wceq 1623    e. wcel 1684    C_ wss 3152   {cpr 3641   CCcc 8735   RRcr 8736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-sn 3646  df-pr 3647
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