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Theorem cp 7023
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 7017 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 2495 . . 3  |-  z  e.  _V
21cplem2 7022 . 2  |-  E. w A. x  e.  z 
( { y  | 
ph }  =/=  (/)  ->  ( { y  |  ph }  i^i  w )  =/=  (/) )
3 abn0 3106 . . . . 5  |-  ( {
y  |  ph }  =/=  (/)  <->  E. y ph )
4 elin 2995 . . . . . . . 8  |-  ( y  e.  ( { y  | 
ph }  i^i  w
)  <->  ( y  e. 
{ y  |  ph }  /\  y  e.  w
) )
5 abid 2071 . . . . . . . . 9  |-  ( y  e.  { y  |  ph } 
<-> 
ph )
65anbi1i 672 . . . . . . . 8  |-  ( (
y  e.  { y  |  ph }  /\  y  e.  w )  <->  (
ph  /\  y  e.  w ) )
7 ancom 431 . . . . . . . 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 1491 . . . . . 6  |-  ( E. y  y  e.  ( { y  |  ph }  i^i  w )  <->  E. y
( y  e.  w  /\  ph ) )
10 hbab1 2072 . . . . . . . 8  |-  ( z  e.  { y  |  ph }  ->  A. y  z  e. 
{ y  |  ph } )
11 ax-17 1545 . . . . . . . 8  |-  ( z  e.  w  ->  A. y 
z  e.  w )
1210, 11hbin 3011 . . . . . . 7  |-  ( z  e.  ( { y  | 
ph }  i^i  w
)  ->  A. y 
z  e.  ( { y  |  ph }  i^i  w ) )
1312n0f 3096 . . . . . 6  |-  ( ( { y  |  ph }  i^i  w )  =/=  (/) 
<->  E. y  y  e.  ( { y  | 
ph }  i^i  w
) )
14 df-rex 2296 . . . . . 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 2313 . . 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 1491 . 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 357   E.wex 1455    e. wcel 1538   {cab 2069    =/= wne 2199   A.wral 2291   E.wrex 2292    i^i cin 2809   (/)c0 3088
This theorem is referenced by:  bnd  7024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1451  ax-6 1452  ax-7 1453  ax-gen 1454  ax-8 1540  ax-11 1541  ax-13 1542  ax-14 1543  ax-17 1545  ax-12o 1578  ax-10 1592  ax-9 1598  ax-4 1606  ax-16 1793  ax-ext 2064  ax-rep 3705  ax-sep 3715  ax-nul 3723  ax-pow 3759  ax-pr 3783  ax-un 4075  ax-reg 6769  ax-inf2 6805
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-3or 904  df-3an 905  df-ex 1456  df-sb 1754  df-eu 1976  df-mo 1977  df-clab 2070  df-cleq 2075  df-clel 2078  df-ne 2201  df-ral 2295  df-rex 2296  df-reu 2297  df-rab 2298  df-v 2494  df-sbc 2668  df-csb 2750  df-dif 2813  df-un 2815  df-in 2817  df-ss 2821  df-pss 2823  df-nul 3089  df-if 3199  df-pw 3260  df-sn 3278  df-pr 3279  df-tp 3280  df-op 3281  df-uni 3439  df-int 3473  df-iun 3516  df-iin 3517  df-br 3601  df-opab 3655  df-mpt 3656  df-tr 3688  df-eprel 3870  df-id 3874  df-po 3879  df-so 3880  df-fr 3917  df-we 3919  df-ord 3960  df-on 3961  df-lim 3962  df-suc 3963  df-om 4243  df-xp 4289  df-rel 4290  df-cnv 4291  df-co 4292  df-dm 4293  df-rn 4294  df-res 4295  df-ima 4296  df-fun 4297  df-fn 4298  df-f 4299  df-f1 4300  df-fo 4301  df-f1o 4302  df-fv 4303  df-recs 5835  df-rdg 5870  df-r1 6899  df-rank 6900
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