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

Theorem iscard 8345
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 8314 . . 3  |-  ( card `  A )  e.  On
2 eleq1 2532 . . 3  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 211 . 2  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 cardonle 8327 . . . 4  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
5 eqss 3512 . . . . 5  |-  ( (
card `  A )  =  A  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) )
65baibr 899 . . . 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 4896 . . . . . 6  |-  ( ( A  e.  On  /\  x  e.  A )  ->  x  e.  On )
9 onenon 8319 . . . . . . 7  |-  ( A  e.  On  ->  A  e.  dom  card )
109adantr 465 . . . . . 6  |-  ( ( A  e.  On  /\  x  e.  A )  ->  A  e.  dom  card )
11 cardsdomel 8344 . . . . . 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 2893 . . . 4  |-  ( A  e.  On  ->  ( A. x  e.  A  x  ~<  A  <->  A. x  e.  A  x  e.  ( card `  A )
) )
14 dfss3 3487 . . . 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 1374    e. wcel 1762   A.wral 2807    C_ wss 3469   class class class wbr 4440   Oncon0 4871   dom cdm 4992   ` cfv 5579    ~< csdm 7505   cardccrd 8305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-card 8309
This theorem is referenced by:  cardprclem  8349  cardmin2  8368  infxpenlem  8380  alephsuc2  8450  cardmin  8928  alephreg  8946  pwcfsdom  8947  winalim2  9063  gchina  9066  inar1  9142  r1tskina  9149  gruina  9185
  Copyright terms: Public domain W3C validator