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Theorem cp 6226
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 6220 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.
Assertion
Ref Expression
cp |- E.wA.x e. z (E.yph -> 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 2361 . . 3 |- z e. _V
21cplem2 6225 . 2 |- E.wA.x e. z ({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/))
3 abn0 2939 . . . . 5 |- ({y | ph} =/= (/) <-> E.yph)
4 elin 2835 . . . . . . . 8 |- (y e. ({y | ph} i^i w) <-> (y e. {y | ph} /\ y e. w))
5 abid 1942 . . . . . . . . 9 |- (y e. {y | ph} <-> ph)
65anbi1i 695 . . . . . . . 8 |- ((y e. {y | ph} /\ y e. w) <-> (ph /\ y e. w))
7 ancom 453 . . . . . . . 8 |- ((ph /\ y e. w) <-> (y e. w /\ ph))
84, 6, 73bitri 277 . . . . . . 7 |- (y e. ({y | ph} i^i w) <-> (y e. w /\ ph))
98exbii 1404 . . . . . 6 |- (E.y y e. ({y | ph} i^i w) <-> E.y(y e. w /\ ph))
10 hbab1 1943 . . . . . . . 8 |- (z e. {y | ph} -> A.y z e. {y | ph})
11 ax-17 1465 . . . . . . . 8 |- (z e. w -> A.y z e. w)
1210, 11hbin 2848 . . . . . . 7 |- (z e. ({y | ph} i^i w) -> A.y z e. ({y | ph} i^i w))
1312n0f 2929 . . . . . 6 |- (({y | ph} i^i w) =/= (/) <-> E.y y e. ({y | ph} i^i w))
14 df-rex 2167 . . . . . 6 |- (E.y e. w ph <-> E.y(y e. w /\ ph))
159, 13, 143bitr4i 283 . . . . 5 |- (({y | ph} i^i w) =/= (/) <-> E.y e. w ph)
163, 15imbi12i 331 . . . 4 |- (({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/)) <-> (E.yph -> E.y e. w ph))
1716ralbii 2184 . . 3 |- (A.x e. z ({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/)) <-> A.x e. z (E.yph -> E.y e. w ph))
1817exbii 1404 . 2 |- (E.wA.x e. z ({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/)) <-> E.wA.x e. z (E.yph -> E.y e. w ph))
192, 18mpbi 212 1 |- E.wA.x e. z (E.yph -> E.y e. w ph)
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 377  E.wex 1371   e. wcel 1451  {cab 1940   =/= wne 2071  A.wral 2162  E.wrex 2163   i^i cin 2656  (/)c0 2921
This theorem is referenced by:  bnd 6227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1367  ax-6 1368  ax-7 1369  ax-gen 1370  ax-8 1453  ax-10 1454  ax-11 1455  ax-12 1456  ax-13 1457  ax-14 1458  ax-17 1465  ax-9 1480  ax-4 1486  ax-16 1664  ax-ext 1935  ax-rep 3459  ax-sep 3469  ax-nul 3478  ax-pow 3514  ax-pr 3538  ax-un 3808  ax-reg 6036  ax-inf2 6071
This theorem depends on definitions:  df-bi 185  df-or 378  df-an 379  df-3or 938  df-3an 939  df-ex 1372  df-sb 1626  df-eu 1853  df-mo 1854  df-clab 1941  df-cleq 1946  df-clel 1949  df-ne 2073  df-ral 2166  df-rex 2167  df-rab 2169  df-v 2360  df-sbc 2525  df-csb 2600  df-dif 2660  df-un 2662  df-in 2664  df-ss 2666  df-pss 2668  df-nul 2922  df-if 3023  df-pw 3081  df-sn 3096  df-pr 3097  df-tp 3099  df-op 3100  df-uni 3229  df-int 3263  df-iun 3301  df-iin 3302  df-br 3374  df-opab 3428  df-tr 3443  df-eprel 3621  df-id 3624  df-po 3629  df-so 3643  df-fr 3662  df-we 3678  df-ord 3694  df-on 3695  df-lim 3696  df-suc 3697  df-om 3971  df-xp 4018  df-rel 4019  df-cnv 4020  df-co 4021  df-dm 4022  df-rn 4023  df-res 4024  df-ima 4025  df-fun 4026  df-fn 4027  df-f 4028  df-fv 4032  df-mpt 5068  df-rdg 5356  df-r1 6113  df-rank 6114
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