MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  canth3 Unicode version

Theorem canth3 8178
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 7010 . 2  |-  ( A  e.  V  ->  A  ~<  ~P A )
2 pwexg 4193 . . 3  |-  ( A  e.  V  ->  ~P A  e.  _V )
3 cardsdom 8172 . . 3  |-  ( ( A  e.  V  /\  ~P A  e.  _V )  ->  ( ( card `  A )  e.  (
card `  ~P A )  <-> 
A  ~<  ~P A ) )
42, 3mpdan 651 . 2  |-  ( A  e.  V  ->  (
( card `  A )  e.  ( card `  ~P A )  <->  A  ~<  ~P A ) )
51, 4mpbird 225 1  |-  ( A  e.  V  ->  ( card `  A )  e.  ( card `  ~P A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    e. wcel 1685   _Vcvv 2789   ~Pcpw 3626   class class class wbr 4024   ` cfv 5221    ~< csdm 6857   cardccrd 7563
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-ac2 8084
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-iota 6252  df-riota 6299  df-recs 6383  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-card 7567  df-ac 7738
  Copyright terms: Public domain W3C validator