Theorem cp 5852

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 5846 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

Proof of Theorem cp

Step	Hyp	Ref	Expression
1		visset 2295	. . 3 |- z e. _V
2	1	cplem2 5851	. 2 |- E.wA.x e. z ({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/))
3		abn0 2892	. . . . 5 |- ({y | ph} =/= (/) <-> E.yph)
4		elin 2786	. . . . . . . 8 |- (y e. ({y | ph} i^i w) <-> (y e. {y | ph} /\ y e. w))
5		abid 1873	. . . . . . . . 9 |- (y e. {y | ph} <-> ph)
6	5	anbi1i 539	. . . . . . . 8 |- ((y e. {y | ph} /\ y e. w) <-> (ph /\ y e. w))
7		ancom 482	. . . . . . . 8 |- ((ph /\ y e. w) <-> (y e. w /\ ph))
8	4, 6, 7	3bitri 194	. . . . . . 7 |- (y e. ({y | ph} i^i w) <-> (y e. w /\ ph))
9	8	exbii 1398	. . . . . 6 |- (E.y y e. ({y | ph} i^i w) <-> E.y(y e. w /\ ph))
10		hbab1 1874	. . . . . . . 8 |- (z e. {y | ph} -> A.y z e. {y | ph})
11		ax-17 1317	. . . . . . . 8 |- (z e. w -> A.y z e. w)
12	10, 11	hbin 2800	. . . . . . 7 |- (z e. ({y | ph} i^i w) -> A.y z e. ({y | ph} i^i w))
13	12	ne0f 2883	. . . . . 6 |- (({y | ph} i^i w) =/= (/) <-> E.y y e. ({y | ph} i^i w))
14		df-rex 2110	. . . . . 6 |- (E.y e. w ph <-> E.y(y e. w /\ ph))
15	9, 13, 14	3bitr4i 200	. . . . 5 |- (({y | ph} i^i w) =/= (/) <-> E.y e. w ph)
16	3, 15	imbi12i 205	. . . 4 |- (({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/)) <-> (E.yph -> E.y e. w ph))
17	16	ralbii 2127	. . 3 |- (A.x e. z ({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/)) <-> A.x e. z (E.yph -> E.y e. w ph))
18	17	exbii 1398	. 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))
19	2, 18	mpbi 206	1 |- E.wA.x e. z (E.yph -> E.y e. w ph)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  A.wral 2105  E.wrex 2106   i^i cin 2592  (/)c0 2875

This theorem is referenced by:  bnd 5853

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140  df-r1 5750  df-rank 5751

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