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Theorem iscard2 7625
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 7593 . . 3  |-  ( card `  A )  e.  On
2 eleq1 2356 . . 3  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 202 . 2  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 cardonle 7606 . . . . . 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 3207 . . . . 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 7604 . . . . 5  |-  ( A  e.  On  ->  ( card `  A )  = 
|^| { y  e.  On  |  y  ~~  A }
)
98sseq2d 3219 . . . 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 3894 . . . 4  |-  ( A 
C_  |^| { y  e.  On  |  y  ~~  A }  <->  A. x  e.  {
y  e.  On  | 
y  ~~  A } A  C_  x )
12 breq1 4042 . . . . . . . . 9  |-  ( y  =  x  ->  (
y  ~~  A  <->  x  ~~  A ) )
1312elrab 2936 . . . . . . . 8  |-  ( x  e.  { y  e.  On  |  y  ~~  A }  <->  ( x  e.  On  /\  x  ~~  A ) )
14 ensymb 6925 . . . . . . . . 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 2584 . . . 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 1632    e. wcel 1696   A.wral 2556   {crab 2560    C_ wss 3165   |^|cint 3878   class class class wbr 4039   Oncon0 4408   ` cfv 5271    ~~ cen 6876   cardccrd 7584
This theorem is referenced by:  harcard  7627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-card 7588
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