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Theorem cflem 7867
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 3198 . . 3  |-  A  C_  A
2 ssid 3198 . . . . 5  |-  z  C_  z
3 sseq2 3201 . . . . . 6  |-  ( w  =  z  ->  (
z  C_  w  <->  z  C_  z ) )
43rspcev 2885 . . . . 5  |-  ( ( z  e.  A  /\  z  C_  z )  ->  E. w  e.  A  z  C_  w )
52, 4mpan2 654 . . . 4  |-  ( z  e.  A  ->  E. w  e.  A  z  C_  w )
65rgen 2609 . . 3  |-  A. z  e.  A  E. w  e.  A  z  C_  w
7 sseq1 3200 . . . . 5  |-  ( y  =  A  ->  (
y  C_  A  <->  A  C_  A
) )
8 rexeq 2738 . . . . . 6  |-  ( y  =  A  ->  ( E. w  e.  y 
z  C_  w  <->  E. w  e.  A  z  C_  w ) )
98ralbidv 2564 . . . . 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 693 . . . 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 ) ) )
1110spcegv 2870 . . 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 661 . 2  |-  ( A  e.  V  ->  E. y
( y  C_  A  /\  A. z  e.  A  E. w  e.  y 
z  C_  w )
)
13 fvex 5499 . . . . . 6  |-  ( card `  y )  e.  _V
1413isseti 2795 . . . . 5  |-  E. x  x  =  ( card `  y )
15 19.41v 1843 . . . . 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 886 . . . 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 1570 . . 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 1787 . . 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 1529    = wceq 1624    e. wcel 1685   A.wral 2544   E.wrex 2545    C_ wss 3153   ` cfv 5221   cardccrd 7563
This theorem is referenced by:  cfval  7868  cff  7869  cff1  7879
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-sn 3647  df-pr 3648  df-uni 3829  df-fv 5229
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