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Theorem canth3 8744
Description: Cantor's theorem in terms of cardinals. This theorem tells us that no matter how large 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 7484 . 2  |-  ( A  e.  V  ->  A  ~<  ~P A )
2 pwexg 4495 . . 3  |-  ( A  e.  V  ->  ~P A  e.  _V )
3 cardsdom 8738 . . 3  |-  ( ( A  e.  V  /\  ~P A  e.  _V )  ->  ( ( card `  A )  e.  (
card `  ~P A )  <-> 
A  ~<  ~P A ) )
42, 3mpdan 668 . 2  |-  ( A  e.  V  ->  (
( card `  A )  e.  ( card `  ~P A )  <->  A  ~<  ~P A ) )
51, 4mpbird 232 1  |-  ( A  e.  V  ->  ( card `  A )  e.  ( card `  ~P A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1756   _Vcvv 2991   ~Pcpw 3879   class class class wbr 4311   ` cfv 5437    ~< csdm 7328   cardccrd 8124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-ac2 8651
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-se 4699  df-we 4700  df-ord 4741  df-on 4742  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-isom 5446  df-riota 6071  df-recs 6851  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-card 8128  df-ac 8305
This theorem is referenced by: (None)
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