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Theorem cp 6188
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 6182 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 2368 . . 3 |- z e. _V
21cplem2 6187 . 2 |- E.wA.x e. z ({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/))
3 abn0 2944 . . . . 5 |- ({y | ph} =/= (/) <-> E.yph)
4 elin 2840 . . . . . . . 8 |- (y e. ({y | ph} i^i w) <-> (y e. {y | ph} /\ y e. w))
5 abid 1949 . . . . . . . . 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 1950 . . . . . . . 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 2853 . . . . . . 7 |- (z e. ({y | ph} i^i w) -> A.y z e. ({y | ph} i^i w))
1312n0f 2934 . . . . . 6 |- (({y | ph} i^i w) =/= (/) <-> E.y y e. ({y | ph} i^i w))
14 df-rex 2174 . . . . . 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 2191 . . 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 1947   =/= wne 2078  A.wral 2169  E.wrex 2170   i^i cin 2661  (/)c0 2926
This theorem is referenced by:  bnd 6189
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 1671  ax-ext 1942  ax-rep 3462  ax-sep 3472  ax-nul 3481  ax-pow 3517  ax-pr 3541  ax-un 3811  ax-reg 5998  ax-inf2 6033
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 1633  df-eu 1860  df-mo 1861  df-clab 1948  df-cleq 1953  df-clel 1956  df-ne 2080  df-ral 2173  df-rex 2174  df-rab 2176  df-v 2367  df-sbc 2532  df-csb 2606  df-dif 2665  df-un 2667  df-in 2669  df-ss 2671  df-pss 2673  df-nul 2927  df-if 3028  df-pw 3084  df-sn 3099  df-pr 3100  df-tp 3102  df-op 3103  df-uni 3232  df-int 3266  df-iun 3304  df-iin 3305  df-br 3377  df-opab 3431  df-tr 3446  df-eprel 3624  df-id 3627  df-po 3632  df-so 3646  df-fr 3665  df-we 3681  df-ord 3697  df-on 3698  df-lim 3699  df-suc 3700  df-om 3967  df-xp 4014  df-rel 4015  df-cnv 4016  df-co 4017  df-dm 4018  df-rn 4019  df-res 4020  df-ima 4021  df-fun 4022  df-fn 4023  df-f 4024  df-fv 4028  df-mpt 5061  df-rdg 5320  df-r1 6075  df-rank 6076
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