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Theorem iscard2 7852
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 7820 . . 3  |-  ( card `  A )  e.  On
2 eleq1 2495 . . 3  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 203 . 2  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 cardonle 7833 . . . . . 6  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
54biantrurd 495 . . . . 5  |-  ( A  e.  On  ->  ( A  C_  ( card `  A
)  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) ) )
6 eqss 3355 . . . . 5  |-  ( (
card `  A )  =  A  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) )
75, 6syl6rbbr 256 . . . 4  |-  ( A  e.  On  ->  (
( card `  A )  =  A  <->  A  C_  ( card `  A ) ) )
8 oncardval 7831 . . . . 5  |-  ( A  e.  On  ->  ( card `  A )  = 
|^| { y  e.  On  |  y  ~~  A }
)
98sseq2d 3368 . . . 4  |-  ( A  e.  On  ->  ( A  C_  ( card `  A
)  <->  A  C_  |^| { y  e.  On  |  y 
~~  A } ) )
107, 9bitrd 245 . . 3  |-  ( A  e.  On  ->  (
( card `  A )  =  A  <->  A  C_  |^| { y  e.  On  |  y 
~~  A } ) )
11 ssint 4058 . . . 4  |-  ( A 
C_  |^| { y  e.  On  |  y  ~~  A }  <->  A. x  e.  {
y  e.  On  | 
y  ~~  A } A  C_  x )
12 breq1 4207 . . . . . . . . 9  |-  ( y  =  x  ->  (
y  ~~  A  <->  x  ~~  A ) )
1312elrab 3084 . . . . . . . 8  |-  ( x  e.  { y  e.  On  |  y  ~~  A }  <->  ( x  e.  On  /\  x  ~~  A ) )
14 ensymb 7146 . . . . . . . . 9  |-  ( x 
~~  A  <->  A  ~~  x )
1514anbi2i 676 . . . . . . . 8  |-  ( ( x  e.  On  /\  x  ~~  A )  <->  ( x  e.  On  /\  A  ~~  x ) )
1613, 15bitri 241 . . . . . . 7  |-  ( x  e.  { y  e.  On  |  y  ~~  A }  <->  ( x  e.  On  /\  A  ~~  x ) )
1716imbi1i 316 . . . . . 6  |-  ( ( x  e.  { y  e.  On  |  y 
~~  A }  ->  A 
C_  x )  <->  ( (
x  e.  On  /\  A  ~~  x )  ->  A  C_  x ) )
18 impexp 434 . . . . . 6  |-  ( ( ( x  e.  On  /\  A  ~~  x )  ->  A  C_  x
)  <->  ( x  e.  On  ->  ( A  ~~  x  ->  A  C_  x ) ) )
1917, 18bitri 241 . . . . 5  |-  ( ( x  e.  { y  e.  On  |  y 
~~  A }  ->  A 
C_  x )  <->  ( x  e.  On  ->  ( A  ~~  x  ->  A  C_  x ) ) )
2019ralbii2 2725 . . . 4  |-  ( A. x  e.  { y  e.  On  |  y  ~~  A } A  C_  x  <->  A. x  e.  On  ( A  ~~  x  ->  A  C_  x ) )
2111, 20bitri 241 . . 3  |-  ( A 
C_  |^| { y  e.  On  |  y  ~~  A }  <->  A. x  e.  On  ( A  ~~  x  ->  A  C_  x ) )
2210, 21syl6bb 253 . 2  |-  ( A  e.  On  ->  (
( card `  A )  =  A  <->  A. x  e.  On  ( A  ~~  x  ->  A  C_  x ) ) )
233, 22biadan2 624 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701    C_ wss 3312   |^|cint 4042   class class class wbr 4204   Oncon0 4573   ` cfv 5445    ~~ cen 7097   cardccrd 7811
This theorem is referenced by:  harcard  7854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-er 6896  df-en 7101  df-card 7815
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