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Theorem iscard2 7609
Description: Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225. (Contributed by Mario Carneiro, 15-Jan-2013.)
Assertion
Ref Expression
iscard2  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  On  ( A  ~~  x  ->  A  C_  x
) ) )
Distinct variable group:    x, A

Proof of Theorem iscard2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cardon 7577 . . 3  |-  ( card `  A )  e.  On
2 eleq1 2343 . . 3  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 202 . 2  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 cardonle 7590 . . . . . 6  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
54biantrurd 494 . . . . 5  |-  ( A  e.  On  ->  ( A  C_  ( card `  A
)  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) ) )
6 eqss 3194 . . . . 5  |-  ( (
card `  A )  =  A  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) )
75, 6syl6rbbr 255 . . . 4  |-  ( A  e.  On  ->  (
( card `  A )  =  A  <->  A  C_  ( card `  A ) ) )
8 oncardval 7588 . . . . 5  |-  ( A  e.  On  ->  ( card `  A )  = 
|^| { y  e.  On  |  y  ~~  A }
)
98sseq2d 3206 . . . 4  |-  ( A  e.  On  ->  ( A  C_  ( card `  A
)  <->  A  C_  |^| { y  e.  On  |  y 
~~  A } ) )
107, 9bitrd 244 . . 3  |-  ( A  e.  On  ->  (
( card `  A )  =  A  <->  A  C_  |^| { y  e.  On  |  y 
~~  A } ) )
11 ssint 3878 . . . 4  |-  ( A 
C_  |^| { y  e.  On  |  y  ~~  A }  <->  A. x  e.  {
y  e.  On  | 
y  ~~  A } A  C_  x )
12 breq1 4026 . . . . . . . . 9  |-  ( y  =  x  ->  (
y  ~~  A  <->  x  ~~  A ) )
1312elrab 2923 . . . . . . . 8  |-  ( x  e.  { y  e.  On  |  y  ~~  A }  <->  ( x  e.  On  /\  x  ~~  A ) )
14 ensymb 6909 . . . . . . . . 9  |-  ( x 
~~  A  <->  A  ~~  x )
1514anbi2i 675 . . . . . . . 8  |-  ( ( x  e.  On  /\  x  ~~  A )  <->  ( x  e.  On  /\  A  ~~  x ) )
1613, 15bitri 240 . . . . . . 7  |-  ( x  e.  { y  e.  On  |  y  ~~  A }  <->  ( x  e.  On  /\  A  ~~  x ) )
1716imbi1i 315 . . . . . 6  |-  ( ( x  e.  { y  e.  On  |  y 
~~  A }  ->  A 
C_  x )  <->  ( (
x  e.  On  /\  A  ~~  x )  ->  A  C_  x ) )
18 impexp 433 . . . . . 6  |-  ( ( ( x  e.  On  /\  A  ~~  x )  ->  A  C_  x
)  <->  ( x  e.  On  ->  ( A  ~~  x  ->  A  C_  x ) ) )
1917, 18bitri 240 . . . . 5  |-  ( ( x  e.  { y  e.  On  |  y 
~~  A }  ->  A 
C_  x )  <->  ( x  e.  On  ->  ( A  ~~  x  ->  A  C_  x ) ) )
2019ralbii2 2571 . . . 4  |-  ( A. x  e.  { y  e.  On  |  y  ~~  A } A  C_  x  <->  A. x  e.  On  ( A  ~~  x  ->  A  C_  x ) )
2111, 20bitri 240 . . 3  |-  ( A 
C_  |^| { y  e.  On  |  y  ~~  A }  <->  A. x  e.  On  ( A  ~~  x  ->  A  C_  x ) )
2210, 21syl6bb 252 . 2  |-  ( A  e.  On  ->  (
( card `  A )  =  A  <->  A. x  e.  On  ( A  ~~  x  ->  A  C_  x ) ) )
233, 22biadan2 623 1  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  On  ( A  ~~  x  ->  A  C_  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    C_ wss 3152   |^|cint 3862   class class class wbr 4023   Oncon0 4392   ` cfv 5255    ~~ cen 6860   cardccrd 7568
This theorem is referenced by:  harcard  7611
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-rab 2552  df-v 2790  df-sbc 2992  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-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-we 4354  df-ord 4395  df-on 4396  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-er 6660  df-en 6864  df-card 7572
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