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Theorem cp 6233
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 6227 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 2346 . . 3 |- z e. _V
21cplem2 6232 . 2 |- E.wA.x e. z ({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/))
3 abn0 2924 . . . . 5 |- ({y | ph} =/= (/) <-> E.yph)
4 elin 2820 . . . . . . . 8 |- (y e. ({y | ph} i^i w) <-> (y e. {y | ph} /\ y e. w))
5 abid 1927 . . . . . . . . 9 |- (y e. {y | ph} <-> ph)
65anbi1i 681 . . . . . . . 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 1389 . . . . . 6 |- (E.y y e. ({y | ph} i^i w) <-> E.y(y e. w /\ ph))
10 hbab1 1928 . . . . . . . 8 |- (z e. {y | ph} -> A.y z e. {y | ph})
11 ax-17 1450 . . . . . . . 8 |- (z e. w -> A.y z e. w)
1210, 11hbin 2833 . . . . . . 7 |- (z e. ({y | ph} i^i w) -> A.y z e. ({y | ph} i^i w))
1312n0f 2914 . . . . . 6 |- (({y | ph} i^i w) =/= (/) <-> E.y y e. ({y | ph} i^i w))
14 df-rex 2152 . . . . . 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.yph -> E.y e. w ph))
1716ralbii 2169 . . 3 |- (A.x e. z ({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/)) <-> A.x e. z (E.yph -> E.y e. w ph))
1817exbii 1389 . 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 197 1 |- E.wA.x e. z (E.yph -> E.y e. w ph)
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 361  E.wex 1355   e. wcel 1436  {cab 1925   =/= wne 2056  A.wral 2147  E.wrex 2148   i^i cin 2641  (/)c0 2906
This theorem is referenced by:  bnd 6234
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1351  ax-6 1352  ax-7 1353  ax-gen 1354  ax-8 1438  ax-10 1439  ax-11 1440  ax-12 1441  ax-13 1442  ax-14 1443  ax-17 1450  ax-9 1465  ax-4 1471  ax-16 1649  ax-ext 1920  ax-rep 3449  ax-sep 3459  ax-nul 3468  ax-pow 3504  ax-pr 3528  ax-un 3800  ax-reg 6044  ax-inf2 6079
This theorem depends on definitions:  df-bi 175  df-or 362  df-an 363  df-3or 922  df-3an 923  df-ex 1356  df-sb 1611  df-eu 1838  df-mo 1839  df-clab 1926  df-cleq 1931  df-clel 1934  df-ne 2058  df-ral 2151  df-rex 2152  df-rab 2154  df-v 2345  df-sbc 2510  df-csb 2585  df-dif 2645  df-un 2647  df-in 2649  df-ss 2651  df-pss 2653  df-nul 2907  df-if 3010  df-pw 3068  df-sn 3085  df-pr 3086  df-tp 3087  df-op 3088  df-uni 3219  df-int 3253  df-iun 3291  df-iin 3292  df-br 3364  df-opab 3418  df-tr 3433  df-eprel 3613  df-id 3616  df-po 3621  df-so 3635  df-fr 3654  df-we 3670  df-ord 3686  df-on 3687  df-lim 3688  df-suc 3689  df-om 3963  df-xp 4010  df-rel 4011  df-cnv 4012  df-co 4013  df-dm 4014  df-rn 4015  df-res 4016  df-ima 4017  df-fun 4018  df-fn 4019  df-f 4020  df-fv 4024  df-mpt 5065  df-rdg 5364  df-r1 6120  df-rank 6121
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