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Theorem card2inf 7523
Description: The definition cardval2 7878 has the curious property that for non-numerable sets (for which ndmfv 5755 yields  (/)), it still evaluates to a non-empty 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 4215 . . . . 5  |-  ( x  =  (/)  ->  ( x 
~<  A  <->  (/)  ~<  A )
)
2 breq1 4215 . . . . 5  |-  ( x  =  n  ->  (
x  ~<  A  <->  n  ~<  A ) )
3 breq1 4215 . . . . 5  |-  ( x  =  suc  n  -> 
( x  ~<  A  <->  suc  n  ~<  A ) )
4 0elon 4634 . . . . . . . 8  |-  (/)  e.  On
5 breq1 4215 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( y 
~~  A  <->  (/)  ~~  A
) )
65rspcev 3052 . . . . . . . 8  |-  ( (
(/)  e.  On  /\  (/)  ~~  A
)  ->  E. y  e.  On  y  ~~  A
)
74, 6mpan 652 . . . . . . 7  |-  ( (/)  ~~  A  ->  E. y  e.  On  y  ~~  A
)
87con3i 129 . . . . . 6  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  -.  (/)  ~~  A )
9 card2inf.1 . . . . . . . 8  |-  A  e. 
_V
1090dom 7237 . . . . . . 7  |-  (/)  ~<_  A
11 brsdom 7130 . . . . . . 7  |-  ( (/)  ~<  A 
<->  ( (/)  ~<_  A  /\  -.  (/)  ~~  A )
)
1210, 11mpbiran 885 . . . . . 6  |-  ( (/)  ~<  A 
<->  -.  (/)  ~~  A )
138, 12sylibr 204 . . . . 5  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  (/)  ~<  A )
14 sucdom2 7303 . . . . . . . 8  |-  ( n 
~<  A  ->  suc  n  ~<_  A )
1514ad2antll 710 . . . . . . 7  |-  ( ( n  e.  om  /\  ( -.  E. y  e.  On  y  ~~  A  /\  n  ~<  A ) )  ->  suc  n  ~<_  A )
16 nnon 4851 . . . . . . . . . 10  |-  ( n  e.  om  ->  n  e.  On )
17 suceloni 4793 . . . . . . . . . 10  |-  ( n  e.  On  ->  suc  n  e.  On )
18 breq1 4215 . . . . . . . . . . . 12  |-  ( y  =  suc  n  -> 
( y  ~~  A  <->  suc  n  ~~  A ) )
1918rspcev 3052 . . . . . . . . . . 11  |-  ( ( suc  n  e.  On  /\ 
suc  n  ~~  A
)  ->  E. y  e.  On  y  ~~  A
)
2019ex 424 . . . . . . . . . 10  |-  ( suc  n  e.  On  ->  ( suc  n  ~~  A  ->  E. y  e.  On  y  ~~  A ) )
2116, 17, 203syl 19 . . . . . . . . 9  |-  ( n  e.  om  ->  ( suc  n  ~~  A  ->  E. y  e.  On  y  ~~  A ) )
2221con3and 429 . . . . . . . 8  |-  ( ( n  e.  om  /\  -.  E. y  e.  On  y  ~~  A )  ->  -.  suc  n  ~~  A
)
2322adantrr 698 . . . . . . 7  |-  ( ( n  e.  om  /\  ( -.  E. y  e.  On  y  ~~  A  /\  n  ~<  A ) )  ->  -.  suc  n  ~~  A )
24 brsdom 7130 . . . . . . 7  |-  ( suc  n  ~<  A  <->  ( suc  n  ~<_  A  /\  -.  suc  n  ~~  A ) )
2515, 23, 24sylanbrc 646 . . . . . 6  |-  ( ( n  e.  om  /\  ( -.  E. y  e.  On  y  ~~  A  /\  n  ~<  A ) )  ->  suc  n  ~<  A )
2625exp32 589 . . . . 5  |-  ( n  e.  om  ->  ( -.  E. y  e.  On  y  ~~  A  ->  (
n  ~<  A  ->  suc  n  ~<  A ) ) )
271, 2, 3, 13, 26finds2 4873 . . . 4  |-  ( x  e.  om  ->  ( -.  E. y  e.  On  y  ~~  A  ->  x  ~<  A ) )
2827com12 29 . . 3  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  (
x  e.  om  ->  x 
~<  A ) )
2928ralrimiv 2788 . 2  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  A. x  e.  om  x  ~<  A )
30 omsson 4849 . . 3  |-  om  C_  On
31 ssrab 3421 . . 3  |-  ( om  C_  { x  e.  On  |  x  ~<  A }  <->  ( om  C_  On  /\  A. x  e.  om  x  ~<  A ) )
3230, 31mpbiran 885 . 2  |-  ( om  C_  { x  e.  On  |  x  ~<  A }  <->  A. x  e.  om  x  ~<  A )
3329, 32sylibr 204 1  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  om  C_  { x  e.  On  |  x  ~<  A } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    e. wcel 1725   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2956    C_ wss 3320   (/)c0 3628   class class class wbr 4212   Oncon0 4581   suc csuc 4583   omcom 4845    ~~ cen 7106    ~<_ cdom 7107    ~< csdm 7108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-1o 6724  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112
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