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Theorem cp 6334
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 6328 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 visset 2502 . . 3 |- z e. _V
21cplem2 6333 . 2 |- E.wA.x e. z ({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/))
3 abn0 3083 . . . . 5 |- ({y | ph} =/= (/) <-> E.yph)
4 elin 2978 . . . . . . . 8 |- (y e. ({y | ph} i^i w) <-> (y e. {y | ph} /\ y e. w))
5 abid 2081 . . . . . . . . 9 |- (y e. {y | ph} <-> ph)
65anbi1i 787 . . . . . . . 8 |- ((y e. {y | ph} /\ y e. w) <-> (ph /\ y e. w))
7 ancom 501 . . . . . . . 8 |- ((ph /\ y e. w) <-> (y e. w /\ ph))
84, 6, 73bitri 314 . . . . . . 7 |- (y e. ({y | ph} i^i w) <-> (y e. w /\ ph))
98exbii 1553 . . . . . 6 |- (E.y y e. ({y | ph} i^i w) <-> E.y(y e. w /\ ph))
10 hbab1 2082 . . . . . . . 8 |- (z e. {y | ph} -> A.y z e. {y | ph})
11 ax-17 1608 . . . . . . . 8 |- (z e. w -> A.y z e. w)
1210, 11hbin 2991 . . . . . . 7 |- (z e. ({y | ph} i^i w) -> A.y z e. ({y | ph} i^i w))
1312n0f 3073 . . . . . 6 |- (({y | ph} i^i w) =/= (/) <-> E.y y e. ({y | ph} i^i w))
14 df-rex 2315 . . . . . 6 |- (E.y e. w ph <-> E.y(y e. w /\ ph))
159, 13, 143bitr4i 320 . . . . 5 |- (({y | ph} i^i w) =/= (/) <-> E.y e. w ph)
163, 15imbi12i 371 . . . 4 |- (({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/)) <-> (E.yph -> E.y e. w ph))
1716ralbii 2332 . . 3 |- (A.x e. z ({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/)) <-> A.x e. z (E.yph -> E.y e. w ph))
1817exbii 1553 . 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 237 1 |- E.wA.x e. z (E.yph -> E.y e. w ph)
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 418  E.wex 1520   e. wcel 1594  {cab 2079   =/= wne 2218  A.wral 2310  E.wrex 2311   i^i cin 2800  (/)c0 3065
This theorem is referenced by:  bnd 6335
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1516  ax-6 1517  ax-7 1518  ax-gen 1519  ax-8 1596  ax-10 1597  ax-11 1598  ax-12 1599  ax-13 1600  ax-14 1601  ax-17 1608  ax-9 1620  ax-4 1626  ax-16 1803  ax-ext 2074  ax-rep 3599  ax-sep 3609  ax-nul 3619  ax-pow 3655  ax-pr 3679  ax-un 3947  ax-reg 6102  ax-inf2 6137
This theorem depends on definitions:  df-bi 210  df-or 419  df-an 420  df-3or 1038  df-3an 1039  df-ex 1521  df-sb 1765  df-eu 1992  df-mo 1993  df-clab 2080  df-cleq 2085  df-clel 2088  df-ne 2220  df-ral 2314  df-rex 2315  df-rab 2317  df-v 2501  df-sbc 2671  df-csb 2745  df-dif 2804  df-un 2806  df-in 2808  df-ss 2810  df-pss 2812  df-nul 3066  df-if 3166  df-pw 3222  df-sn 3237  df-pr 3238  df-tp 3240  df-op 3241  df-uni 3365  df-int 3399  df-iun 3437  df-iin 3438  df-br 3510  df-opab 3568  df-tr 3583  df-eprel 3762  df-id 3765  df-po 3770  df-so 3782  df-fr 3800  df-we 3816  df-ord 3832  df-on 3833  df-lim 3834  df-suc 3835  df-om 4104  df-xp 4151  df-rel 4152  df-cnv 4153  df-co 4154  df-dm 4155  df-rn 4156  df-res 4157  df-ima 4158  df-fun 4159  df-fn 4160  df-f 4161  df-fv 4165  df-mpt 5202  df-rdg 5460  df-r1 6182  df-rank 6183
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