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Theorem card2inf 8073
Description: The definition cardval2 8427 has the curious property that for non-numerable sets (for which ndmfv 5902 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 4423 . . . . 5  |-  ( x  =  (/)  ->  ( x 
~<  A  <->  (/)  ~<  A )
)
2 breq1 4423 . . . . 5  |-  ( x  =  n  ->  (
x  ~<  A  <->  n  ~<  A ) )
3 breq1 4423 . . . . 5  |-  ( x  =  suc  n  -> 
( x  ~<  A  <->  suc  n  ~<  A ) )
4 0elon 5492 . . . . . . . 8  |-  (/)  e.  On
5 breq1 4423 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( y 
~~  A  <->  (/)  ~~  A
) )
65rspcev 3182 . . . . . . . 8  |-  ( (
(/)  e.  On  /\  (/)  ~~  A
)  ->  E. y  e.  On  y  ~~  A
)
74, 6mpan 674 . . . . . . 7  |-  ( (/)  ~~  A  ->  E. y  e.  On  y  ~~  A
)
87con3i 140 . . . . . 6  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  -.  (/)  ~~  A )
9 card2inf.1 . . . . . . . 8  |-  A  e. 
_V
1090dom 7705 . . . . . . 7  |-  (/)  ~<_  A
11 brsdom 7596 . . . . . . 7  |-  ( (/)  ~<  A 
<->  ( (/)  ~<_  A  /\  -.  (/)  ~~  A )
)
1210, 11mpbiran 926 . . . . . 6  |-  ( (/)  ~<  A 
<->  -.  (/)  ~~  A )
138, 12sylibr 215 . . . . 5  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  (/)  ~<  A )
14 sucdom2 7771 . . . . . . . 8  |-  ( n 
~<  A  ->  suc  n  ~<_  A )
1514ad2antll 733 . . . . . . 7  |-  ( ( n  e.  om  /\  ( -.  E. y  e.  On  y  ~~  A  /\  n  ~<  A ) )  ->  suc  n  ~<_  A )
16 nnon 6709 . . . . . . . . . 10  |-  ( n  e.  om  ->  n  e.  On )
17 suceloni 6651 . . . . . . . . . 10  |-  ( n  e.  On  ->  suc  n  e.  On )
18 breq1 4423 . . . . . . . . . . . 12  |-  ( y  =  suc  n  -> 
( y  ~~  A  <->  suc  n  ~~  A ) )
1918rspcev 3182 . . . . . . . . . . 11  |-  ( ( suc  n  e.  On  /\ 
suc  n  ~~  A
)  ->  E. y  e.  On  y  ~~  A
)
2019ex 435 . . . . . . . . . 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 442 . . . . . . . 8  |-  ( ( n  e.  om  /\  -.  E. y  e.  On  y  ~~  A )  ->  -.  suc  n  ~~  A
)
2322adantrr 721 . . . . . . 7  |-  ( ( n  e.  om  /\  ( -.  E. y  e.  On  y  ~~  A  /\  n  ~<  A ) )  ->  -.  suc  n  ~~  A )
24 brsdom 7596 . . . . . . 7  |-  ( suc  n  ~<  A  <->  ( suc  n  ~<_  A  /\  -.  suc  n  ~~  A ) )
2515, 23, 24sylanbrc 668 . . . . . 6  |-  ( ( n  e.  om  /\  ( -.  E. y  e.  On  y  ~~  A  /\  n  ~<  A ) )  ->  suc  n  ~<  A )
2625exp32 608 . . . . 5  |-  ( n  e.  om  ->  ( -.  E. y  e.  On  y  ~~  A  ->  (
n  ~<  A  ->  suc  n  ~<  A ) ) )
271, 2, 3, 13, 26finds2 6732 . . . 4  |-  ( x  e.  om  ->  ( -.  E. y  e.  On  y  ~~  A  ->  x  ~<  A ) )
2827com12 32 . . 3  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  (
x  e.  om  ->  x 
~<  A ) )
2928ralrimiv 2837 . 2  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  A. x  e.  om  x  ~<  A )
30 omsson 6707 . . 3  |-  om  C_  On
31 ssrab 3539 . . 3  |-  ( om  C_  { x  e.  On  |  x  ~<  A }  <->  ( om  C_  On  /\  A. x  e.  om  x  ~<  A ) )
3230, 31mpbiran 926 . 2  |-  ( om  C_  { x  e.  On  |  x  ~<  A }  <->  A. x  e.  om  x  ~<  A )
3329, 32sylibr 215 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 370    e. wcel 1868   A.wral 2775   E.wrex 2776   {crab 2779   _Vcvv 3081    C_ wss 3436   (/)c0 3761   class class class wbr 4420   Oncon0 5439   suc csuc 5441   omcom 6703    ~~ cen 7571    ~<_ cdom 7572    ~< csdm 7573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594
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 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-om 6704  df-1o 7187  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577
This theorem is referenced by: (None)
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