Theorem cp 4694

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 4688 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 1804	. . 3 |- z e. V
2	1	cplem2 4693	. 2 |- E.wA.x e. z ({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/))
3		abn0 2280	. . . . 5 |- ({y | ph} =/= (/) <-> E.yph)
4		elin 2197	. . . . . . . 8 |- (y e. ({y | ph} i^i w) <-> (y e. {y | ph} /\ y e. w))
5		abid 1458	. . . . . . . . 9 |- (y e. {y | ph} <-> ph)
6	5	anbi1i 480	. . . . . . . 8 |- ((y e. {y | ph} /\ y e. w) <-> (ph /\ y e. w))
7		ancom 435	. . . . . . . 8 |- ((ph /\ y e. w) <-> (y e. w /\ ph))
8	4, 6, 7	3bitr 177	. . . . . . 7 |- (y e. ({y | ph} i^i w) <-> (y e. w /\ ph))
9	8	exbii 1047	. . . . . 6 |- (E.y y e. ({y | ph} i^i w) <-> E.y(y e. w /\ ph))
10		hbab1 1459	. . . . . . . 8 |- (z e. {y | ph} -> A.y z e. {y | ph})
11		ax-17 968	. . . . . . . 8 |- (z e. w -> A.y z e. w)
12	10, 11	hbin 2210	. . . . . . 7 |- (z e. ({y | ph} i^i w) -> A.y z e. ({y | ph} i^i w))
13	12	ne0f 2277	. . . . . 6 |- (({y | ph} i^i w) =/= (/) <-> E.y y e. ({y | ph} i^i w))
14		df-rex 1642	. . . . . 6 |- (E.y e. w ph <-> E.y(y e. w /\ ph))
15	9, 13, 14	3bitr4 183	. . . . 5 |- (({y | ph} i^i w) =/= (/) <-> E.y e. w ph)
16	3, 15	imbi12i 188	. . . 4 |- (({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/)) <-> (E.yph -> E.y e. w ph))
17	16	ralbii 1659	. . 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 1047	. 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 189	1 |- E.wA.x e. z (E.yph -> E.y e. w ph)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 955  E.wex 977  {cab 1456   =/= wne 1577  A.wral 1637  E.wrex 1638   i^i cin 2036  (/)c0 2270

This theorem is referenced by:  bnd 4695

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-iin 2559  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-r1 4615  df-rank 4616

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