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Theorem iscard 8359
Description: Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.)
Assertion
Ref Expression
iscard  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  A  x  ~<  A ) )
Distinct variable group:    x, A

Proof of Theorem iscard
StepHypRef Expression
1 cardon 8328 . . 3  |-  ( card `  A )  e.  On
2 eleq1 2515 . . 3  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 211 . 2  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 cardonle 8341 . . . 4  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
5 eqss 3504 . . . . 5  |-  ( (
card `  A )  =  A  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) )
65baibr 904 . . . 4  |-  ( (
card `  A )  C_  A  ->  ( A  C_  ( card `  A
)  <->  ( card `  A
)  =  A ) )
74, 6syl 16 . . 3  |-  ( A  e.  On  ->  ( A  C_  ( card `  A
)  <->  ( card `  A
)  =  A ) )
8 onelon 4893 . . . . . 6  |-  ( ( A  e.  On  /\  x  e.  A )  ->  x  e.  On )
9 onenon 8333 . . . . . . 7  |-  ( A  e.  On  ->  A  e.  dom  card )
109adantr 465 . . . . . 6  |-  ( ( A  e.  On  /\  x  e.  A )  ->  A  e.  dom  card )
11 cardsdomel 8358 . . . . . 6  |-  ( ( x  e.  On  /\  A  e.  dom  card )  ->  ( x  ~<  A  <->  x  e.  ( card `  A )
) )
128, 10, 11syl2anc 661 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  A )  ->  ( x  ~<  A  <->  x  e.  ( card `  A )
) )
1312ralbidva 2879 . . . 4  |-  ( A  e.  On  ->  ( A. x  e.  A  x  ~<  A  <->  A. x  e.  A  x  e.  ( card `  A )
) )
14 dfss3 3479 . . . 4  |-  ( A 
C_  ( card `  A
)  <->  A. x  e.  A  x  e.  ( card `  A ) )
1513, 14syl6rbbr 264 . . 3  |-  ( A  e.  On  ->  ( A  C_  ( card `  A
)  <->  A. x  e.  A  x  ~<  A ) )
167, 15bitr3d 255 . 2  |-  ( A  e.  On  ->  (
( card `  A )  =  A  <->  A. x  e.  A  x  ~<  A ) )
173, 16biadan2 642 1  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  A  x  ~<  A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793    C_ wss 3461   class class class wbr 4437   Oncon0 4868   dom cdm 4989   ` cfv 5578    ~< csdm 7517   cardccrd 8319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-card 8323
This theorem is referenced by:  cardprclem  8363  cardmin2  8382  infxpenlem  8394  alephsuc2  8464  cardmin  8942  alephreg  8960  pwcfsdom  8961  winalim2  9077  gchina  9080  inar1  9156  r1tskina  9163  gruina  9199
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