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Theorem ruc 12618
Description: The set of natural numbers is strictly dominated by the set of real numbers, i.e. the real numbers are uncountable. The proof consists of lemmas ruclem1 12606 through ruclem13 12617 and this final piece. Our proof is based on the proof of Theorem 5.18 of [Truss] p. 114. See ruclem13 12617 for the function existence version of this theorem. For an informal discussion of this proof, see http://us.metamath.org/mpeuni/mmcomplex.html#uncountable. For an alternate proof see rucALT 12605. (Contributed by NM, 13-Oct-2004.)
Assertion
Ref Expression
ruc  |-  NN  ~<  RR

Proof of Theorem ruc
StepHypRef Expression
1 reex 8918 . . 3  |-  RR  e.  _V
2 nnssre 9840 . . 3  |-  NN  C_  RR
3 ssdomg 6995 . . 3  |-  ( RR  e.  _V  ->  ( NN  C_  RR  ->  NN  ~<_  RR ) )
41, 2, 3mp2 17 . 2  |-  NN  ~<_  RR
5 ruclem13 12617 . . . . 5  |-  -.  f : NN -onto-> RR
6 f1ofo 5562 . . . . 5  |-  ( f : NN -1-1-onto-> RR  ->  f : NN -onto-> RR )
75, 6mto 167 . . . 4  |-  -.  f : NN -1-1-onto-> RR
87nex 1555 . . 3  |-  -.  E. f  f : NN -1-1-onto-> RR
9 bren 6959 . . 3  |-  ( NN 
~~  RR  <->  E. f  f : NN -1-1-onto-> RR )
108, 9mtbir 290 . 2  |-  -.  NN  ~~  RR
11 brsdom 6972 . 2  |-  ( NN 
~<  RR  <->  ( NN  ~<_  RR  /\  -.  NN  ~~  RR ) )
124, 10, 11mpbir2an 886 1  |-  NN  ~<  RR
Colors of variables: wff set class
Syntax hints:   -. wn 3   E.wex 1541    e. wcel 1710   _Vcvv 2864    C_ wss 3228   class class class wbr 4104   -onto->wfo 5335   -1-1-onto->wf1o 5336    ~~ cen 6948    ~<_ cdom 6949    ~< csdm 6950   RRcr 8826   NNcn 9836
This theorem is referenced by:  resdomq  12619  aleph1re  12620  aleph1irr  12621
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-sup 7284  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-n0 10058  df-z 10117  df-uz 10323  df-fz 10875  df-seq 11139
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