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Theorem canth3 8183
Description: Cantor's theorem in terms of cardinals. This theorem tells us that no matter how largei a cardinal number is, there is a still larger cardinal number. Theorem 18.12 of [Monk1] p. 133. (Contributed by NM, 5-Nov-2003.)
Assertion
Ref Expression
canth3  |-  ( A  e.  V  ->  ( card `  A )  e.  ( card `  ~P A ) )

Proof of Theorem canth3
StepHypRef Expression
1 canth2g 7015 . 2  |-  ( A  e.  V  ->  A  ~<  ~P A )
2 pwexg 4194 . . 3  |-  ( A  e.  V  ->  ~P A  e.  _V )
3 cardsdom 8177 . . 3  |-  ( ( A  e.  V  /\  ~P A  e.  _V )  ->  ( ( card `  A )  e.  (
card `  ~P A )  <-> 
A  ~<  ~P A ) )
42, 3mpdan 649 . 2  |-  ( A  e.  V  ->  (
( card `  A )  e.  ( card `  ~P A )  <->  A  ~<  ~P A ) )
51, 4mpbird 223 1  |-  ( A  e.  V  ->  ( card `  A )  e.  ( card `  ~P A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684   _Vcvv 2788   ~Pcpw 3625   class class class wbr 4023   ` cfv 5255    ~< csdm 6862   cardccrd 7568
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-ac2 8089
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 6304  df-recs 6388  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-card 7572  df-ac 7743
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