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Theorem cp 7053
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 7047 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 2483 . . 3  |-  z  e. 
_V
21cplem2 7052 . 2  |-  E. w A. x  e.  z 
( { y  | 
ph }  =/=  (/)  ->  ( { y  |  ph }  i^i  w )  =/=  (/) )
3 abn0 3095 . . . . 5  |-  ( { y  |  ph }  =/=  (/)  <->  E. y ph )
4 elin 2983 . . . . . . . 8  |-  ( y  e.  ( { y  |  ph }  i^i  w )  <->  ( y  e.  { y  |  ph }  /\  y  e.  w
) )
5 abid 2058 . . . . . . . . 9  |-  ( y  e.  { y  | 
ph }  <->  ph )
65anbi1i 669 . . . . . . . 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 1484 . . . . . 6  |-  ( E. y  y  e.  ( { y  |  ph }  i^i  w )  <->  E. y
( y  e.  w  /\  ph ) )
10 hbab1 2059 . . . . . . . 8  |-  ( z  e.  { y  | 
ph }  ->  A. y 
z  e.  { y  |  ph } )
11 ax-17 1533 . . . . . . . 8  |-  ( z  e.  w  ->  A. y 
z  e.  w )
1210, 11hbin 2999 . . . . . . 7  |-  ( z  e.  ( { y  |  ph }  i^i  w )  ->  A. y 
z  e.  ( { y  |  ph }  i^i  w ) )
1312n0f 3085 . . . . . 6  |-  ( ( { y  |  ph }  i^i  w )  =/=  (/) 
<->  E. y  y  e.  ( { y  | 
ph }  i^i  w
) )
14 df-rex 2284 . . . . . 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 2301 . . 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 1484 . 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 1450    e. wcel 1526   {cab 2056    =/= wne 2187   A.wral 2279   E.wrex 2280    i^i cin 2797   (/)c0 3077
This theorem is referenced by:  bnd  7054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1446  ax-6 1447  ax-7 1448  ax-gen 1449  ax-8 1528  ax-11 1529  ax-13 1530  ax-14 1531  ax-17 1533  ax-12o 1567  ax-10 1581  ax-9 1587  ax-4 1594  ax-16 1780  ax-ext 2051  ax-rep 3701  ax-sep 3711  ax-nul 3719  ax-pow 3755  ax-pr 3779  ax-un 4071  ax-reg 6799  ax-inf2 6835
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 897  df-3an 898  df-ex 1451  df-sb 1741  df-eu 1963  df-mo 1964  df-clab 2057  df-cleq 2062  df-clel 2065  df-ne 2189  df-ral 2283  df-rex 2284  df-reu 2285  df-rab 2286  df-v 2482  df-sbc 2656  df-csb 2738  df-dif 2801  df-un 2803  df-in 2805  df-ss 2809  df-pss 2811  df-nul 3078  df-if 3187  df-pw 3248  df-sn 3266  df-pr 3267  df-tp 3268  df-op 3269  df-uni 3435  df-int 3469  df-iun 3512  df-iin 3513  df-br 3597  df-opab 3651  df-mpt 3652  df-tr 3684  df-eprel 3866  df-id 3870  df-po 3875  df-so 3876  df-fr 3913  df-we 3915  df-ord 3956  df-on 3957  df-lim 3958  df-suc 3959  df-om 4234  df-xp 4280  df-rel 4281  df-cnv 4282  df-co 4283  df-dm 4284  df-rn 4285  df-res 4286  df-ima 4287  df-fun 4288  df-fn 4289  df-f 4290  df-f1 4291  df-fo 4292  df-f1o 4293  df-fv 4294  df-recs 5853  df-rdg 5888  df-r1 6929  df-rank 6930
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