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Theorem cp 6227
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 6221 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 2369 . . 3 |- z e. _V
21cplem2 6226 . 2 |- E.wA.x e. z ({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/))
3 abn0 2945 . . . . 5 |- ({y | ph} =/= (/) <-> E.yph)
4 elin 2841 . . . . . . . 8 |- (y e. ({y | ph} i^i w) <-> (y e. {y | ph} /\ y e. w))
5 abid 1950 . . . . . . . . 9 |- (y e. {y | ph} <-> ph)
65anbi1i 704 . . . . . . . 8 |- ((y e. {y | ph} /\ y e. w) <-> (ph /\ y e. w))
7 ancom 459 . . . . . . . 8 |- ((ph /\ y e. w) <-> (y e. w /\ ph))
84, 6, 73bitri 282 . . . . . . 7 |- (y e. ({y | ph} i^i w) <-> (y e. w /\ ph))
98exbii 1414 . . . . . 6 |- (E.y y e. ({y | ph} i^i w) <-> E.y(y e. w /\ ph))
10 hbab1 1951 . . . . . . . 8 |- (z e. {y | ph} -> A.y z e. {y | ph})
11 ax-17 1473 . . . . . . . 8 |- (z e. w -> A.y z e. w)
1210, 11hbin 2854 . . . . . . 7 |- (z e. ({y | ph} i^i w) -> A.y z e. ({y | ph} i^i w))
1312n0f 2935 . . . . . 6 |- (({y | ph} i^i w) =/= (/) <-> E.y y e. ({y | ph} i^i w))
14 df-rex 2175 . . . . . 6 |- (E.y e. w ph <-> E.y(y e. w /\ ph))
159, 13, 143bitr4i 288 . . . . 5 |- (({y | ph} i^i w) =/= (/) <-> E.y e. w ph)
163, 15imbi12i 336 . . . 4 |- (({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/)) <-> (E.yph -> E.y e. w ph))
1716ralbii 2192 . . 3 |- (A.x e. z ({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/)) <-> A.x e. z (E.yph -> E.y e. w ph))
1817exbii 1414 . 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 217 1 |- E.wA.x e. z (E.yph -> E.y e. w ph)
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 382  E.wex 1380   e. wcel 1459  {cab 1948   =/= wne 2079  A.wral 2170  E.wrex 2171   i^i cin 2662  (/)c0 2927
This theorem is referenced by:  bnd 6228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1376  ax-6 1377  ax-7 1378  ax-gen 1379  ax-8 1461  ax-10 1462  ax-11 1463  ax-12 1464  ax-13 1465  ax-14 1466  ax-17 1473  ax-9 1488  ax-4 1494  ax-16 1672  ax-ext 1943  ax-rep 3465  ax-sep 3475  ax-nul 3484  ax-pow 3520  ax-pr 3544  ax-un 3814  ax-reg 6037  ax-inf2 6072
This theorem depends on definitions:  df-bi 190  df-or 383  df-an 384  df-3or 947  df-3an 948  df-ex 1381  df-sb 1634  df-eu 1861  df-mo 1862  df-clab 1949  df-cleq 1954  df-clel 1957  df-ne 2081  df-ral 2174  df-rex 2175  df-rab 2177  df-v 2368  df-sbc 2533  df-csb 2607  df-dif 2666  df-un 2668  df-in 2670  df-ss 2672  df-pss 2674  df-nul 2928  df-if 3029  df-pw 3087  df-sn 3102  df-pr 3103  df-tp 3105  df-op 3106  df-uni 3235  df-int 3269  df-iun 3307  df-iin 3308  df-br 3380  df-opab 3434  df-tr 3449  df-eprel 3627  df-id 3630  df-po 3635  df-so 3649  df-fr 3668  df-we 3684  df-ord 3700  df-on 3701  df-lim 3702  df-suc 3703  df-om 3975  df-xp 4022  df-rel 4023  df-cnv 4024  df-co 4025  df-dm 4026  df-rn 4027  df-res 4028  df-ima 4029  df-fun 4030  df-fn 4031  df-f 4032  df-fv 4036  df-mpt 5072  df-rdg 5359  df-r1 6114  df-rank 6115
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