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Theorem cflem 7826
Description: A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set  A. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
cflem  |-  ( A  e.  V  ->  E. x E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) )
Distinct variable group:    x, y, z, w, A
Allowed substitution hints:    V( x, y, z, w)

Proof of Theorem cflem
StepHypRef Expression
1 ssid 3158 . . 3  |-  A  C_  A
2 ssid 3158 . . . . 5  |-  z  C_  z
3 sseq2 3161 . . . . . 6  |-  ( w  =  z  ->  (
z  C_  w  <->  z  C_  z ) )
43rcla4ev 2852 . . . . 5  |-  ( ( z  e.  A  /\  z  C_  z )  ->  E. w  e.  A  z  C_  w )
52, 4mpan2 655 . . . 4  |-  ( z  e.  A  ->  E. w  e.  A  z  C_  w )
65rgen 2581 . . 3  |-  A. z  e.  A  E. w  e.  A  z  C_  w
7 sseq1 3160 . . . . 5  |-  ( y  =  A  ->  (
y  C_  A  <->  A  C_  A
) )
8 rexeq 2709 . . . . . 6  |-  ( y  =  A  ->  ( E. w  e.  y 
z  C_  w  <->  E. w  e.  A  z  C_  w ) )
98ralbidv 2536 . . . . 5  |-  ( y  =  A  ->  ( A. z  e.  A  E. w  e.  y 
z  C_  w  <->  A. z  e.  A  E. w  e.  A  z  C_  w ) )
107, 9anbi12d 694 . . . 4  |-  ( y  =  A  ->  (
( y  C_  A  /\  A. z  e.  A  E. w  e.  y 
z  C_  w )  <->  ( A  C_  A  /\  A. z  e.  A  E. w  e.  A  z  C_  w ) ) )
1110cla4egv 2837 . . 3  |-  ( A  e.  V  ->  (
( A  C_  A  /\  A. z  e.  A  E. w  e.  A  z  C_  w )  ->  E. y ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w
) ) )
121, 6, 11mp2ani 662 . 2  |-  ( A  e.  V  ->  E. y
( y  C_  A  /\  A. z  e.  A  E. w  e.  y 
z  C_  w )
)
13 fvex 5458 . . . . . 6  |-  ( card `  y )  e.  _V
1413isseti 2763 . . . . 5  |-  E. x  x  =  ( card `  y )
15 19.41v 2035 . . . . 5  |-  ( E. x ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) )  <->  ( E. x  x  =  ( card `  y )  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y 
z  C_  w )
) )
1614, 15mpbiran 889 . . . 4  |-  ( E. x ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) )  <->  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) )
1716exbii 1580 . . 3  |-  ( E. y E. x ( x  =  ( card `  y )  /\  (
y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) )  <->  E. y
( y  C_  A  /\  A. z  e.  A  E. w  e.  y 
z  C_  w )
)
18 excom 1765 . . 3  |-  ( E. y E. x ( x  =  ( card `  y )  /\  (
y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) )  <->  E. x E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) )
1917, 18bitr3i 244 . 2  |-  ( E. y ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w
)  <->  E. x E. y
( x  =  (
card `  y )  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y 
z  C_  w )
) )
2012, 19sylib 190 1  |-  ( A  e.  V  ->  E. x E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   A.wral 2516   E.wrex 2517    C_ wss 3113   ` cfv 4659   cardccrd 7522
This theorem is referenced by:  cfval  7827  cff  7828  cff1  7838
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-sn 3606  df-pr 3607  df-uni 3788  df-fv 4675
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