Theorem cp 6091

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 6085 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 2541	. . 3 |- z e. _V
2	1	cplem2 6090	. 2 |- E.wA.x e. z ({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/))
3		abn0 3099	. . . . 5 |- ({y | ph} =/= (/) <-> E.yph)
4		elin 2999	. . . . . . . 8 |- (y e. ({y | ph} i^i w) <-> (y e. {y | ph} /\ y e. w))
5		abid 2130	. . . . . . . . 9 |- (y e. {y | ph} <-> ph)
6	5	anbi1i 709	. . . . . . . 8 |- ((y e. {y | ph} /\ y e. w) <-> (ph /\ y e. w))
7		ancom 414	. . . . . . . 8 |- ((ph /\ y e. w) <-> (y e. w /\ ph))
8	4, 6, 7	3bitri 289	. . . . . . 7 |- (y e. ({y | ph} i^i w) <-> (y e. w /\ ph))
9	8	exbii 1687	. . . . . 6 |- (E.y y e. ({y | ph} i^i w) <-> E.y(y e. w /\ ph))
10		hbab1 2131	. . . . . . . 8 |- (z e. {y | ph} -> A.y z e. {y | ph})
11		ax-17 1605	. . . . . . . 8 |- (z e. w -> A.y z e. w)
12	10, 11	hbin 3012	. . . . . . 7 |- (z e. ({y | ph} i^i w) -> A.y z e. ({y | ph} i^i w))
13	12	n0f 3090	. . . . . 6 |- (({y | ph} i^i w) =/= (/) <-> E.y y e. ({y | ph} i^i w))
14		df-rex 2360	. . . . . 6 |- (E.y e. w ph <-> E.y(y e. w /\ ph))
15	9, 13, 14	3bitr4i 295	. . . . 5 |- (({y | ph} i^i w) =/= (/) <-> E.y e. w ph)
16	3, 15	imbi12i 301	. . . 4 |- (({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/)) <-> (E.yph -> E.y e. w ph))
17	16	ralbii 2377	. . 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 1687	. 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 272	1 |- E.wA.x e. z (E.yph -> E.y e. w ph)

Syntax hints:   -> wi 3   /\ wa 337   e. wcel 1588  E.wex 1615  {cab 2128   =/= wne 2266  A.wral 2355  E.wrex 2356   i^i cin 2826  (/)c0 3082

This theorem is referenced by:  bnd 6092

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1592  ax-gen 1593  ax-8 1594  ax-9 1595  ax-10 1596  ax-11 1597  ax-12 1598  ax-13 1599  ax-14 1600  ax-17 1605  ax-4 1608  ax-5o 1610  ax-6o 1613  ax-9o 1763  ax-10o 1781  ax-16 1854  ax-11o 1864  ax-ext 2123  ax-rep 3596  ax-sep 3606  ax-nul 3613  ax-pow 3649  ax-pr 3687  ax-un 3929  ax-reg 5928  ax-inf2 5964

This theorem depends on definitions:  df-bi 220  df-or 338  df-an 339  df-3or 1103  df-3an 1104  df-ex 1616  df-sb 1816  df-eu 2041  df-mo 2042  df-clab 2129  df-cleq 2134  df-clel 2137  df-ne 2268  df-ral 2359  df-rex 2360  df-rab 2362  df-v 2540  df-sbc 2700  df-csb 2774  df-dif 2830  df-un 2832  df-in 2834  df-ss 2836  df-pss 2838  df-nul 3083  df-if 3181  df-pw 3229  df-sn 3242  df-pr 3243  df-tp 3245  df-op 3246  df-uni 3367  df-int 3401  df-iun 3438  df-iin 3439  df-br 3508  df-opab 3566  df-tr 3580  df-eprel 3744  df-id 3747  df-po 3752  df-so 3764  df-fr 3782  df-we 3798  df-ord 3814  df-on 3815  df-lim 3816  df-suc 3817  df-om 4086  df-xp 4133  df-rel 4134  df-cnv 4135  df-co 4136  df-dm 4137  df-rn 4138  df-res 4139  df-ima 4140  df-fun 4141  df-fn 4142  df-fv 4147  df-rdg 5304  df-r1 5986  df-rank 5987

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