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Theorem iscard2 8418
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 8386 . . 3  |-  ( card `  A )  e.  On
2 eleq1 2495 . . 3  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 214 . 2  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 cardonle 8399 . . . . . 6  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
54biantrurd 510 . . . . 5  |-  ( A  e.  On  ->  ( A  C_  ( card `  A
)  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) ) )
6 eqss 3479 . . . . 5  |-  ( (
card `  A )  =  A  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) )
75, 6syl6rbbr 267 . . . 4  |-  ( A  e.  On  ->  (
( card `  A )  =  A  <->  A  C_  ( card `  A ) ) )
8 oncardval 8397 . . . . 5  |-  ( A  e.  On  ->  ( card `  A )  = 
|^| { y  e.  On  |  y  ~~  A }
)
98sseq2d 3492 . . . 4  |-  ( A  e.  On  ->  ( A  C_  ( card `  A
)  <->  A  C_  |^| { y  e.  On  |  y 
~~  A } ) )
107, 9bitrd 256 . . 3  |-  ( A  e.  On  ->  (
( card `  A )  =  A  <->  A  C_  |^| { y  e.  On  |  y 
~~  A } ) )
11 ssint 4271 . . . 4  |-  ( A 
C_  |^| { y  e.  On  |  y  ~~  A }  <->  A. x  e.  {
y  e.  On  | 
y  ~~  A } A  C_  x )
12 breq1 4426 . . . . . . . . 9  |-  ( y  =  x  ->  (
y  ~~  A  <->  x  ~~  A ) )
1312elrab 3228 . . . . . . . 8  |-  ( x  e.  { y  e.  On  |  y  ~~  A }  <->  ( x  e.  On  /\  x  ~~  A ) )
14 ensymb 7627 . . . . . . . . 9  |-  ( x 
~~  A  <->  A  ~~  x )
1514anbi2i 698 . . . . . . . 8  |-  ( ( x  e.  On  /\  x  ~~  A )  <->  ( x  e.  On  /\  A  ~~  x ) )
1613, 15bitri 252 . . . . . . 7  |-  ( x  e.  { y  e.  On  |  y  ~~  A }  <->  ( x  e.  On  /\  A  ~~  x ) )
1716imbi1i 326 . . . . . 6  |-  ( ( x  e.  { y  e.  On  |  y 
~~  A }  ->  A 
C_  x )  <->  ( (
x  e.  On  /\  A  ~~  x )  ->  A  C_  x ) )
18 impexp 447 . . . . . 6  |-  ( ( ( x  e.  On  /\  A  ~~  x )  ->  A  C_  x
)  <->  ( x  e.  On  ->  ( A  ~~  x  ->  A  C_  x ) ) )
1917, 18bitri 252 . . . . 5  |-  ( ( x  e.  { y  e.  On  |  y 
~~  A }  ->  A 
C_  x )  <->  ( x  e.  On  ->  ( A  ~~  x  ->  A  C_  x ) ) )
2019ralbii2 2851 . . . 4  |-  ( A. x  e.  { y  e.  On  |  y  ~~  A } A  C_  x  <->  A. x  e.  On  ( A  ~~  x  ->  A  C_  x ) )
2111, 20bitri 252 . . 3  |-  ( A 
C_  |^| { y  e.  On  |  y  ~~  A }  <->  A. x  e.  On  ( A  ~~  x  ->  A  C_  x ) )
2210, 21syl6bb 264 . 2  |-  ( A  e.  On  ->  (
( card `  A )  =  A  <->  A. x  e.  On  ( A  ~~  x  ->  A  C_  x ) ) )
233, 22biadan2 646 1  |-  ( (
card `  A )  =  A  <->  ( A  e.  On  /\  A. x  e.  On  ( A  ~~  x  ->  A  C_  x
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2771   {crab 2775    C_ wss 3436   |^|cint 4255   class class class wbr 4423   Oncon0 5442   ` cfv 5601    ~~ cen 7577   cardccrd 8377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ord 5445  df-on 5446  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-er 7374  df-en 7581  df-card 8381
This theorem is referenced by:  harcard  8420
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