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Theorem iscard2 8428
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 8396 . . 3  |-  ( card `  A )  e.  On
2 eleq1 2537 . . 3  |-  ( (
card `  A )  =  A  ->  ( (
card `  A )  e.  On  <->  A  e.  On ) )
31, 2mpbii 216 . 2  |-  ( (
card `  A )  =  A  ->  A  e.  On )
4 cardonle 8409 . . . . . 6  |-  ( A  e.  On  ->  ( card `  A )  C_  A )
54biantrurd 516 . . . . 5  |-  ( A  e.  On  ->  ( A  C_  ( card `  A
)  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) ) )
6 eqss 3433 . . . . 5  |-  ( (
card `  A )  =  A  <->  ( ( card `  A )  C_  A  /\  A  C_  ( card `  A ) ) )
75, 6syl6rbbr 272 . . . 4  |-  ( A  e.  On  ->  (
( card `  A )  =  A  <->  A  C_  ( card `  A ) ) )
8 oncardval 8407 . . . . 5  |-  ( A  e.  On  ->  ( card `  A )  = 
|^| { y  e.  On  |  y  ~~  A }
)
98sseq2d 3446 . . . 4  |-  ( A  e.  On  ->  ( A  C_  ( card `  A
)  <->  A  C_  |^| { y  e.  On  |  y 
~~  A } ) )
107, 9bitrd 261 . . 3  |-  ( A  e.  On  ->  (
( card `  A )  =  A  <->  A  C_  |^| { y  e.  On  |  y 
~~  A } ) )
11 ssint 4242 . . . 4  |-  ( A 
C_  |^| { y  e.  On  |  y  ~~  A }  <->  A. x  e.  {
y  e.  On  | 
y  ~~  A } A  C_  x )
12 breq1 4398 . . . . . . . . 9  |-  ( y  =  x  ->  (
y  ~~  A  <->  x  ~~  A ) )
1312elrab 3184 . . . . . . . 8  |-  ( x  e.  { y  e.  On  |  y  ~~  A }  <->  ( x  e.  On  /\  x  ~~  A ) )
14 ensymb 7635 . . . . . . . . 9  |-  ( x 
~~  A  <->  A  ~~  x )
1514anbi2i 708 . . . . . . . 8  |-  ( ( x  e.  On  /\  x  ~~  A )  <->  ( x  e.  On  /\  A  ~~  x ) )
1613, 15bitri 257 . . . . . . 7  |-  ( x  e.  { y  e.  On  |  y  ~~  A }  <->  ( x  e.  On  /\  A  ~~  x ) )
1716imbi1i 332 . . . . . 6  |-  ( ( x  e.  { y  e.  On  |  y 
~~  A }  ->  A 
C_  x )  <->  ( (
x  e.  On  /\  A  ~~  x )  ->  A  C_  x ) )
18 impexp 453 . . . . . 6  |-  ( ( ( x  e.  On  /\  A  ~~  x )  ->  A  C_  x
)  <->  ( x  e.  On  ->  ( A  ~~  x  ->  A  C_  x ) ) )
1917, 18bitri 257 . . . . 5  |-  ( ( x  e.  { y  e.  On  |  y 
~~  A }  ->  A 
C_  x )  <->  ( x  e.  On  ->  ( A  ~~  x  ->  A  C_  x ) ) )
2019ralbii2 2821 . . . 4  |-  ( A. x  e.  { y  e.  On  |  y  ~~  A } A  C_  x  <->  A. x  e.  On  ( A  ~~  x  ->  A  C_  x ) )
2111, 20bitri 257 . . 3  |-  ( A 
C_  |^| { y  e.  On  |  y  ~~  A }  <->  A. x  e.  On  ( A  ~~  x  ->  A  C_  x ) )
2210, 21syl6bb 269 . 2  |-  ( A  e.  On  ->  (
( card `  A )  =  A  <->  A. x  e.  On  ( A  ~~  x  ->  A  C_  x ) ) )
233, 22biadan2 654 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 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760    C_ wss 3390   |^|cint 4226   class class class wbr 4395   Oncon0 5430   ` cfv 5589    ~~ cen 7584   cardccrd 8387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-er 7381  df-en 7588  df-card 8391
This theorem is referenced by:  harcard  8430
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