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Theorem cp 7039
Description: Collection Principle. This remarkable theorem scheme is in effect a very strong generalization of the Axiom of Replacement. The proof makes use of Scott's trick scottex 7033 that collapses a proper class into a set of minimum rank. The wff  ph can be thought of as  ph ( x ,  y ). Scheme "Collection Principle" of [Jech] p. 72. (Contributed by NM, 17-Oct-2003.)
Assertion
Ref Expression
cp  |-  E. w A. x  e.  z 
( E. y ph  ->  E. y  e.  w  ph )
Distinct variable groups:    ph, z, w   
x, y, z, w
Allowed substitution hints:    ph( x, y)

Proof of Theorem cp
StepHypRef Expression
1 vex 2476 . . 3  |-  z  e. 
_V
21cplem2 7038 . 2  |-  E. w A. x  e.  z 
( { y  | 
ph }  =/=  (/)  ->  ( { y  |  ph }  i^i  w )  =/=  (/) )
3 abn0 3088 . . . . 5  |-  ( { y  |  ph }  =/=  (/)  <->  E. y ph )
4 elin 2976 . . . . . . . 8  |-  ( y  e.  ( { y  |  ph }  i^i  w )  <->  ( y  e.  { y  |  ph }  /\  y  e.  w
) )
5 abid 2051 . . . . . . . . 9  |-  ( y  e.  { y  | 
ph }  <->  ph )
65anbi1i 668 . . . . . . . 8  |-  ( ( y  e.  { y  |  ph }  /\  y  e.  w )  <->  (
ph  /\  y  e.  w ) )
7 ancom 430 . . . . . . . 8  |-  ( (
ph  /\  y  e.  w )  <->  ( y  e.  w  /\  ph )
)
84, 6, 73bitri 260 . . . . . . 7  |-  ( y  e.  ( { y  |  ph }  i^i  w )  <->  ( y  e.  w  /\  ph )
)
98exbii 1478 . . . . . 6  |-  ( E. y  y  e.  ( { y  |  ph }  i^i  w )  <->  E. y
( y  e.  w  /\  ph ) )
10 hbab1 2052 . . . . . . . 8  |-  ( z  e.  { y  | 
ph }  ->  A. y 
z  e.  { y  |  ph } )
11 ax-17 1527 . . . . . . . 8  |-  ( z  e.  w  ->  A. y 
z  e.  w )
1210, 11hbin 2992 . . . . . . 7  |-  ( z  e.  ( { y  |  ph }  i^i  w )  ->  A. y 
z  e.  ( { y  |  ph }  i^i  w ) )
1312n0f 3078 . . . . . 6  |-  ( ( { y  |  ph }  i^i  w )  =/=  (/) 
<->  E. y  y  e.  ( { y  | 
ph }  i^i  w
) )
14 df-rex 2277 . . . . . 6  |-  ( E. y  e.  w  ph  <->  E. y ( y  e.  w  /\  ph )
)
159, 13, 143bitr4i 266 . . . . 5  |-  ( ( { y  |  ph }  i^i  w )  =/=  (/) 
<->  E. y  e.  w  ph )
163, 15imbi12i 314 . . . 4  |-  ( ( { y  |  ph }  =/=  (/)  ->  ( {
y  |  ph }  i^i  w )  =/=  (/) )  <->  ( E. y ph  ->  E. y  e.  w  ph ) )
1716ralbii 2294 . . 3  |-  ( A. x  e.  z  ( { y  |  ph }  =/=  (/)  ->  ( {
y  |  ph }  i^i  w )  =/=  (/) )  <->  A. x  e.  z  ( E. y ph  ->  E. y  e.  w  ph ) )
1817exbii 1478 . 2  |-  ( E. w A. x  e.  z  ( { y  |  ph }  =/=  (/) 
->  ( { y  | 
ph }  i^i  w
)  =/=  (/) )  <->  E. w A. x  e.  z 
( E. y ph  ->  E. y  e.  w  ph ) )
192, 18mpbi 197 1  |-  E. w A. x  e.  z 
( E. y ph  ->  E. y  e.  w  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 356   E.wex 1444    e. wcel 1520   {cab 2049    =/= wne 2180   A.wral 2272   E.wrex 2273    i^i cin 2790   (/)c0 3070
This theorem is referenced by:  bnd  7040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1440  ax-6 1441  ax-7 1442  ax-gen 1443  ax-8 1522  ax-11 1523  ax-13 1524  ax-14 1525  ax-17 1527  ax-12o 1560  ax-10 1574  ax-9 1580  ax-4 1587  ax-16 1773  ax-ext 2044  ax-rep 3689  ax-sep 3699  ax-nul 3707  ax-pow 3743  ax-pr 3767  ax-un 4059  ax-reg 6785  ax-inf2 6821
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 895  df-3an 896  df-ex 1445  df-sb 1734  df-eu 1956  df-mo 1957  df-clab 2050  df-cleq 2055  df-clel 2058  df-ne 2182  df-ral 2276  df-rex 2277  df-reu 2278  df-rab 2279  df-v 2475  df-sbc 2649  df-csb 2731  df-dif 2794  df-un 2796  df-in 2798  df-ss 2802  df-pss 2804  df-nul 3071  df-if 3180  df-pw 3241  df-sn 3259  df-pr 3260  df-tp 3261  df-op 3262  df-uni 3423  df-int 3457  df-iun 3500  df-iin 3501  df-br 3585  df-opab 3639  df-mpt 3640  df-tr 3672  df-eprel 3854  df-id 3858  df-po 3863  df-so 3864  df-fr 3901  df-we 3903  df-ord 3944  df-on 3945  df-lim 3946  df-suc 3947  df-om 4222  df-xp 4268  df-rel 4269  df-cnv 4270  df-co 4271  df-dm 4272  df-rn 4273  df-res 4274  df-ima 4275  df-fun 4276  df-fn 4277  df-f 4278  df-f1 4279  df-fo 4280  df-f1o 4281  df-fv 4282  df-recs 5839  df-rdg 5874  df-r1 6915  df-rank 6916
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