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Theorem card2inf 8075
Description: The definition cardval2 8430 has the curious property that for non-numerable sets (for which ndmfv 5894 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 4408 . . . . 5  |-  ( x  =  (/)  ->  ( x 
~<  A  <->  (/)  ~<  A )
)
2 breq1 4408 . . . . 5  |-  ( x  =  n  ->  (
x  ~<  A  <->  n  ~<  A ) )
3 breq1 4408 . . . . 5  |-  ( x  =  suc  n  -> 
( x  ~<  A  <->  suc  n  ~<  A ) )
4 0elon 5479 . . . . . . . 8  |-  (/)  e.  On
5 breq1 4408 . . . . . . . . 9  |-  ( y  =  (/)  ->  ( y 
~~  A  <->  (/)  ~~  A
) )
65rspcev 3152 . . . . . . . 8  |-  ( (
(/)  e.  On  /\  (/)  ~~  A
)  ->  E. y  e.  On  y  ~~  A
)
74, 6mpan 677 . . . . . . 7  |-  ( (/)  ~~  A  ->  E. y  e.  On  y  ~~  A
)
87con3i 141 . . . . . 6  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  -.  (/)  ~~  A )
9 card2inf.1 . . . . . . . 8  |-  A  e. 
_V
1090dom 7707 . . . . . . 7  |-  (/)  ~<_  A
11 brsdom 7597 . . . . . . 7  |-  ( (/)  ~<  A 
<->  ( (/)  ~<_  A  /\  -.  (/)  ~~  A )
)
1210, 11mpbiran 930 . . . . . 6  |-  ( (/)  ~<  A 
<->  -.  (/)  ~~  A )
138, 12sylibr 216 . . . . 5  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  (/)  ~<  A )
14 sucdom2 7773 . . . . . . . 8  |-  ( n 
~<  A  ->  suc  n  ~<_  A )
1514ad2antll 736 . . . . . . 7  |-  ( ( n  e.  om  /\  ( -.  E. y  e.  On  y  ~~  A  /\  n  ~<  A ) )  ->  suc  n  ~<_  A )
16 nnon 6703 . . . . . . . . . 10  |-  ( n  e.  om  ->  n  e.  On )
17 suceloni 6645 . . . . . . . . . 10  |-  ( n  e.  On  ->  suc  n  e.  On )
18 breq1 4408 . . . . . . . . . . . 12  |-  ( y  =  suc  n  -> 
( y  ~~  A  <->  suc  n  ~~  A ) )
1918rspcev 3152 . . . . . . . . . . 11  |-  ( ( suc  n  e.  On  /\ 
suc  n  ~~  A
)  ->  E. y  e.  On  y  ~~  A
)
2019ex 436 . . . . . . . . . 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 443 . . . . . . . 8  |-  ( ( n  e.  om  /\  -.  E. y  e.  On  y  ~~  A )  ->  -.  suc  n  ~~  A
)
2322adantrr 724 . . . . . . 7  |-  ( ( n  e.  om  /\  ( -.  E. y  e.  On  y  ~~  A  /\  n  ~<  A ) )  ->  -.  suc  n  ~~  A )
24 brsdom 7597 . . . . . . 7  |-  ( suc  n  ~<  A  <->  ( suc  n  ~<_  A  /\  -.  suc  n  ~~  A ) )
2515, 23, 24sylanbrc 671 . . . . . 6  |-  ( ( n  e.  om  /\  ( -.  E. y  e.  On  y  ~~  A  /\  n  ~<  A ) )  ->  suc  n  ~<  A )
2625exp32 610 . . . . 5  |-  ( n  e.  om  ->  ( -.  E. y  e.  On  y  ~~  A  ->  (
n  ~<  A  ->  suc  n  ~<  A ) ) )
271, 2, 3, 13, 26finds2 6726 . . . 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 2802 . 2  |-  ( -. 
E. y  e.  On  y  ~~  A  ->  A. x  e.  om  x  ~<  A )
30 omsson 6701 . . 3  |-  om  C_  On
31 ssrab 3509 . . 3  |-  ( om  C_  { x  e.  On  |  x  ~<  A }  <->  ( om  C_  On  /\  A. x  e.  om  x  ~<  A ) )
3230, 31mpbiran 930 . 2  |-  ( om  C_  { x  e.  On  |  x  ~<  A }  <->  A. x  e.  om  x  ~<  A )
3329, 32sylibr 216 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 371    e. wcel 1889   A.wral 2739   E.wrex 2740   {crab 2743   _Vcvv 3047    C_ wss 3406   (/)c0 3733   class class class wbr 4405   Oncon0 5426   suc csuc 5428   omcom 6697    ~~ 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 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-om 6698  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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