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Theorem wuncn 9536
Description: A weak universe containing  om contains the complex number construction. This theorem is construction-dependent in the literal sense, but will also be satisfied by any other reasonable implementation of the complex numbers. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wuncn.1  |-  ( ph  ->  U  e. WUni )
wuncn.2  |-  ( ph  ->  om  e.  U )
Assertion
Ref Expression
wuncn  |-  ( ph  ->  CC  e.  U )

Proof of Theorem wuncn
StepHypRef Expression
1 df-c 9487 . 2  |-  CC  =  ( R.  X.  R. )
2 wuncn.1 . . 3  |-  ( ph  ->  U  e. WUni )
3 df-nr 9423 . . . 4  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
4 df-ni 9239 . . . . . . . . . . . 12  |-  N.  =  ( om  \  { (/) } )
5 wuncn.2 . . . . . . . . . . . . 13  |-  ( ph  ->  om  e.  U )
62, 5wundif 9081 . . . . . . . . . . . 12  |-  ( ph  ->  ( om  \  { (/)
} )  e.  U
)
74, 6syl5eqel 2546 . . . . . . . . . . 11  |-  ( ph  ->  N.  e.  U )
82, 7, 7wunxp 9091 . . . . . . . . . 10  |-  ( ph  ->  ( N.  X.  N. )  e.  U )
9 elpqn 9292 . . . . . . . . . . . 12  |-  ( x  e.  Q.  ->  x  e.  ( N.  X.  N. ) )
109ssriv 3493 . . . . . . . . . . 11  |-  Q.  C_  ( N.  X.  N. )
1110a1i 11 . . . . . . . . . 10  |-  ( ph  ->  Q.  C_  ( N.  X.  N. ) )
122, 8, 11wunss 9079 . . . . . . . . 9  |-  ( ph  ->  Q.  e.  U )
132, 12wunpw 9074 . . . . . . . 8  |-  ( ph  ->  ~P Q.  e.  U
)
14 prpssnq 9357 . . . . . . . . . . . 12  |-  ( x  e.  P.  ->  x  C. 
Q. )
1514pssssd 3587 . . . . . . . . . . 11  |-  ( x  e.  P.  ->  x  C_ 
Q. )
16 selpw 4006 . . . . . . . . . . 11  |-  ( x  e.  ~P Q.  <->  x  C_  Q. )
1715, 16sylibr 212 . . . . . . . . . 10  |-  ( x  e.  P.  ->  x  e.  ~P Q. )
1817ssriv 3493 . . . . . . . . 9  |-  P.  C_  ~P Q.
1918a1i 11 . . . . . . . 8  |-  ( ph  ->  P.  C_  ~P Q. )
202, 13, 19wunss 9079 . . . . . . 7  |-  ( ph  ->  P.  e.  U )
212, 20, 20wunxp 9091 . . . . . 6  |-  ( ph  ->  ( P.  X.  P. )  e.  U )
222, 21wunpw 9074 . . . . 5  |-  ( ph  ->  ~P ( P.  X.  P. )  e.  U
)
23 enrer 9431 . . . . . . 7  |-  ~R  Er  ( P.  X.  P. )
2423a1i 11 . . . . . 6  |-  ( ph  ->  ~R  Er  ( P. 
X.  P. ) )
2524qsss 7364 . . . . 5  |-  ( ph  ->  ( ( P.  X.  P. ) /.  ~R  )  C_ 
~P ( P.  X.  P. ) )
262, 22, 25wunss 9079 . . . 4  |-  ( ph  ->  ( ( P.  X.  P. ) /.  ~R  )  e.  U )
273, 26syl5eqel 2546 . . 3  |-  ( ph  ->  R.  e.  U )
282, 27, 27wunxp 9091 . 2  |-  ( ph  ->  ( R.  X.  R. )  e.  U )
291, 28syl5eqel 2546 1  |-  ( ph  ->  CC  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1823    \ cdif 3458    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016    X. cxp 4986   omcom 6673    Er wer 7300   /.cqs 7302  WUnicwun 9067   N.cnpi 9211   Q.cnq 9219   P.cnp 9226    ~R cer 9231   R.cnr 9232   CCcc 9479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-omul 7127  df-er 7303  df-ec 7305  df-qs 7309  df-wun 9069  df-ni 9239  df-pli 9240  df-mi 9241  df-lti 9242  df-plpq 9275  df-mpq 9276  df-ltpq 9277  df-enq 9278  df-nq 9279  df-erq 9280  df-plq 9281  df-mq 9282  df-1nq 9283  df-rq 9284  df-ltnq 9285  df-np 9348  df-plp 9350  df-ltp 9352  df-enr 9422  df-nr 9423  df-c 9487
This theorem is referenced by:  wunndx  14735
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