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Theorem card2inf 8088
Description: The definition cardval2 8443 has the curious property that for non-numerable sets (for which ndmfv 5903 yields  (/)), it still evaluates to a nonempty set, and indeed it contains  om. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
Hypothesis
Ref Expression
card2inf.1  |-  A  e. 
_V
Assertion
Ref Expression
card2inf  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  om  C_  { x  e.  On  |  x  ~<  A } )
Distinct variable group:    x, A, y

Proof of Theorem card2inf
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 breq1 4398 . . . . 5  |-  ( x  =  (/)  ->  ( x 
~<  A  <->  (/)  ~<  A )
)
2 breq1 4398 . . . . 5  |-  ( x  =  n  ->  (
x  ~<  A  <->  n  ~<  A ) )
3 breq1 4398 . . . . 5  |-  ( x  =  suc  n  -> 
( x  ~<  A  <->  suc  n  ~<  A ) )
4 0elon 5483 . . . . . . . 8  |-  (/)  e.  On
5 breq1 4398 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( y 
~~  A  <->  (/)  ~~  A
) )
65rspcev 3136 . . . . . . . 8  |-  ( (
(/)  e.  On  /\  (/)  ~~  A
)  ->  E. y  e.  On  y  ~~  A
)
74, 6mpan 684 . . . . . . 7  |-  ( (/)  ~~  A  ->  E. y  e.  On  y  ~~  A
)
87con3i 142 . . . . . 6  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  -.  (/)  ~~  A )
9 card2inf.1 . . . . . . . 8  |-  A  e. 
_V
1090dom 7720 . . . . . . 7  |-  (/)  ~<_  A
11 brsdom 7610 . . . . . . 7  |-  ( (/)  ~<  A 
<->  ( (/)  ~<_  A  /\  -.  (/)  ~~  A )
)
1210, 11mpbiran 932 . . . . . 6  |-  ( (/)  ~<  A 
<->  -.  (/)  ~~  A )
138, 12sylibr 217 . . . . 5  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  (/)  ~<  A )
14 sucdom2 7786 . . . . . . . 8  |-  ( n 
~<  A  ->  suc  n  ~<_  A )
1514ad2antll 743 . . . . . . 7  |-  ( ( n  e.  om  /\  ( -.  E. y  e.  On  y  ~~  A  /\  n  ~<  A ) )  ->  suc  n  ~<_  A )
16 nnon 6717 . . . . . . . . . 10  |-  ( n  e.  om  ->  n  e.  On )
17 suceloni 6659 . . . . . . . . . 10  |-  ( n  e.  On  ->  suc  n  e.  On )
18 breq1 4398 . . . . . . . . . . . 12  |-  ( y  =  suc  n  -> 
( y  ~~  A  <->  suc  n  ~~  A ) )
1918rspcev 3136 . . . . . . . . . . 11  |-  ( ( suc  n  e.  On  /\ 
suc  n  ~~  A
)  ->  E. y  e.  On  y  ~~  A
)
2019ex 441 . . . . . . . . . 10  |-  ( suc  n  e.  On  ->  ( suc  n  ~~  A  ->  E. y  e.  On  y  ~~  A ) )
2116, 17, 203syl 18 . . . . . . . . 9  |-  ( n  e.  om  ->  ( suc  n  ~~  A  ->  E. y  e.  On  y  ~~  A ) )
2221con3dimp 448 . . . . . . . 8  |-  ( ( n  e.  om  /\  -.  E. y  e.  On  y  ~~  A )  ->  -.  suc  n  ~~  A
)
2322adantrr 731 . . . . . . 7  |-  ( ( n  e.  om  /\  ( -.  E. y  e.  On  y  ~~  A  /\  n  ~<  A ) )  ->  -.  suc  n  ~~  A )
24 brsdom 7610 . . . . . . 7  |-  ( suc  n  ~<  A  <->  ( suc  n  ~<_  A  /\  -.  suc  n  ~~  A ) )
2515, 23, 24sylanbrc 677 . . . . . 6  |-  ( ( n  e.  om  /\  ( -.  E. y  e.  On  y  ~~  A  /\  n  ~<  A ) )  ->  suc  n  ~<  A )
2625exp32 616 . . . . 5  |-  ( n  e.  om  ->  ( -.  E. y  e.  On  y  ~~  A  ->  (
n  ~<  A  ->  suc  n  ~<  A ) ) )
271, 2, 3, 13, 26finds2 6740 . . . 4  |-  ( x  e.  om  ->  ( -.  E. y  e.  On  y  ~~  A  ->  x  ~<  A ) )
2827com12 31 . . 3  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  (
x  e.  om  ->  x 
~<  A ) )
2928ralrimiv 2808 . 2  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  A. x  e.  om  x  ~<  A )
30 omsson 6715 . . 3  |-  om  C_  On
31 ssrab 3493 . . 3  |-  ( om  C_  { x  e.  On  |  x  ~<  A }  <->  ( om  C_  On  /\  A. x  e.  om  x  ~<  A ) )
3230, 31mpbiran 932 . 2  |-  ( om  C_  { x  e.  On  |  x  ~<  A }  <->  A. x  e.  om  x  ~<  A )
3329, 32sylibr 217 1  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  om  C_  { x  e.  On  |  x  ~<  A } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031    C_ wss 3390   (/)c0 3722   class class class wbr 4395   Oncon0 5430   suc csuc 5432   omcom 6711    ~~ cen 7584    ~<_ cdom 7585    ~< csdm 7586
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-br 4396  df-opab 4455  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-lim 5435  df-suc 5436  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-om 6712  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590
This theorem is referenced by: (None)
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