MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cp Unicode version

Theorem cp 7561
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 7555 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 2791 . . 3  |-  z  e. 
_V
21cplem2 7560 . 2  |-  E. w A. x  e.  z 
( { y  | 
ph }  =/=  (/)  ->  ( { y  |  ph }  i^i  w )  =/=  (/) )
3 abn0 3473 . . . . 5  |-  ( { y  |  ph }  =/=  (/)  <->  E. y ph )
4 elin 3358 . . . . . . . 8  |-  ( y  e.  ( { y  |  ph }  i^i  w )  <->  ( y  e.  { y  |  ph }  /\  y  e.  w
) )
5 abid 2271 . . . . . . . . 9  |-  ( y  e.  { y  | 
ph }  <->  ph )
65anbi1i 676 . . . . . . . 8  |-  ( ( y  e.  { y  |  ph }  /\  y  e.  w )  <->  (
ph  /\  y  e.  w ) )
7 ancom 437 . . . . . . . 8  |-  ( (
ph  /\  y  e.  w )  <->  ( y  e.  w  /\  ph )
)
84, 6, 73bitri 262 . . . . . . 7  |-  ( y  e.  ( { y  |  ph }  i^i  w )  <->  ( y  e.  w  /\  ph )
)
98exbii 1569 . . . . . 6  |-  ( E. y  y  e.  ( { y  |  ph }  i^i  w )  <->  E. y
( y  e.  w  /\  ph ) )
10 nfab1 2421 . . . . . . . 8  |-  F/_ y { y  |  ph }
11 nfcv 2419 . . . . . . . 8  |-  F/_ y
w
1210, 11nfin 3375 . . . . . . 7  |-  F/_ y
( { y  | 
ph }  i^i  w
)
1312n0f 3463 . . . . . 6  |-  ( ( { y  |  ph }  i^i  w )  =/=  (/) 
<->  E. y  y  e.  ( { y  | 
ph }  i^i  w
) )
14 df-rex 2549 . . . . . 6  |-  ( E. y  e.  w  ph  <->  E. y ( y  e.  w  /\  ph )
)
159, 13, 143bitr4i 268 . . . . 5  |-  ( ( { y  |  ph }  i^i  w )  =/=  (/) 
<->  E. y  e.  w  ph )
163, 15imbi12i 316 . . . 4  |-  ( ( { y  |  ph }  =/=  (/)  ->  ( {
y  |  ph }  i^i  w )  =/=  (/) )  <->  ( E. y ph  ->  E. y  e.  w  ph ) )
1716ralbii 2567 . . 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 1569 . 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 199 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 358   E.wex 1528    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544    i^i cin 3151   (/)c0 3455
This theorem is referenced by:  bnd  7562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-r1 7436  df-rank 7437
  Copyright terms: Public domain W3C validator