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Theorem card2inf 7973
Description: The definition cardval2 8363 has the curious property that for non-numerable sets (for which ndmfv 5872 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 4442 . . . . 5  |-  ( x  =  (/)  ->  ( x 
~<  A  <->  (/)  ~<  A )
)
2 breq1 4442 . . . . 5  |-  ( x  =  n  ->  (
x  ~<  A  <->  n  ~<  A ) )
3 breq1 4442 . . . . 5  |-  ( x  =  suc  n  -> 
( x  ~<  A  <->  suc  n  ~<  A ) )
4 0elon 4920 . . . . . . . 8  |-  (/)  e.  On
5 breq1 4442 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( y 
~~  A  <->  (/)  ~~  A
) )
65rspcev 3207 . . . . . . . 8  |-  ( (
(/)  e.  On  /\  (/)  ~~  A
)  ->  E. y  e.  On  y  ~~  A
)
74, 6mpan 668 . . . . . . 7  |-  ( (/)  ~~  A  ->  E. y  e.  On  y  ~~  A
)
87con3i 135 . . . . . 6  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  -.  (/)  ~~  A )
9 card2inf.1 . . . . . . . 8  |-  A  e. 
_V
1090dom 7640 . . . . . . 7  |-  (/)  ~<_  A
11 brsdom 7531 . . . . . . 7  |-  ( (/)  ~<  A 
<->  ( (/)  ~<_  A  /\  -.  (/)  ~~  A )
)
1210, 11mpbiran 916 . . . . . 6  |-  ( (/)  ~<  A 
<->  -.  (/)  ~~  A )
138, 12sylibr 212 . . . . 5  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  (/)  ~<  A )
14 sucdom2 7707 . . . . . . . 8  |-  ( n 
~<  A  ->  suc  n  ~<_  A )
1514ad2antll 726 . . . . . . 7  |-  ( ( n  e.  om  /\  ( -.  E. y  e.  On  y  ~~  A  /\  n  ~<  A ) )  ->  suc  n  ~<_  A )
16 nnon 6679 . . . . . . . . . 10  |-  ( n  e.  om  ->  n  e.  On )
17 suceloni 6621 . . . . . . . . . 10  |-  ( n  e.  On  ->  suc  n  e.  On )
18 breq1 4442 . . . . . . . . . . . 12  |-  ( y  =  suc  n  -> 
( y  ~~  A  <->  suc  n  ~~  A ) )
1918rspcev 3207 . . . . . . . . . . 11  |-  ( ( suc  n  e.  On  /\ 
suc  n  ~~  A
)  ->  E. y  e.  On  y  ~~  A
)
2019ex 432 . . . . . . . . . 10  |-  ( suc  n  e.  On  ->  ( suc  n  ~~  A  ->  E. y  e.  On  y  ~~  A ) )
2116, 17, 203syl 20 . . . . . . . . 9  |-  ( n  e.  om  ->  ( suc  n  ~~  A  ->  E. y  e.  On  y  ~~  A ) )
2221con3dimp 439 . . . . . . . 8  |-  ( ( n  e.  om  /\  -.  E. y  e.  On  y  ~~  A )  ->  -.  suc  n  ~~  A
)
2322adantrr 714 . . . . . . 7  |-  ( ( n  e.  om  /\  ( -.  E. y  e.  On  y  ~~  A  /\  n  ~<  A ) )  ->  -.  suc  n  ~~  A )
24 brsdom 7531 . . . . . . 7  |-  ( suc  n  ~<  A  <->  ( suc  n  ~<_  A  /\  -.  suc  n  ~~  A ) )
2515, 23, 24sylanbrc 662 . . . . . 6  |-  ( ( n  e.  om  /\  ( -.  E. y  e.  On  y  ~~  A  /\  n  ~<  A ) )  ->  suc  n  ~<  A )
2625exp32 603 . . . . 5  |-  ( n  e.  om  ->  ( -.  E. y  e.  On  y  ~~  A  ->  (
n  ~<  A  ->  suc  n  ~<  A ) ) )
271, 2, 3, 13, 26finds2 6701 . . . 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 2866 . 2  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  A. x  e.  om  x  ~<  A )
30 omsson 6677 . . 3  |-  om  C_  On
31 ssrab 3564 . . 3  |-  ( om  C_  { x  e.  On  |  x  ~<  A }  <->  ( om  C_  On  /\  A. x  e.  om  x  ~<  A ) )
3230, 31mpbiran 916 . 2  |-  ( om  C_  { x  e.  On  |  x  ~<  A }  <->  A. x  e.  om  x  ~<  A )
3329, 32sylibr 212 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 367    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106    C_ wss 3461   (/)c0 3783   class class class wbr 4439   Oncon0 4867   suc csuc 4869   omcom 6673    ~~ cen 7506    ~<_ cdom 7507    ~< csdm 7508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-om 6674  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512
This theorem is referenced by: (None)
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