Theorem cp 4646

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 4640 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 1788	. . 3 |- z e. V
2	1	cplem2 4645	. 2 |- E.wA.x e. z ({y | ph} =/= (/) -> ({y | ph} i^i w) =/= (/))
3		abn0 2261	. . . . 5 |- ({y | ph} =/= (/) <-> E.yph)
4		elin 2178	. . . . . . . 8 |- (y e. ({y | ph} i^i w) <-> (y e. {y | ph} /\ y e. w))
5		abid 1442	. . . . . . . . 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 1027	. . . . . 6 |- (E.y y e. ({y | ph} i^i w) <-> E.y(y e. w /\ ph))
10		hbab1 1443	. . . . . . . 8 |- (z e. {y | ph} -> A.y z e. {y | ph})
11		ax-17 1190	. . . . . . . 8 |- (z e. w -> A.y z e. w)
12	10, 11	hbin 2191	. . . . . . 7 |- (z e. ({y | ph} i^i w) -> A.y z e. ({y | ph} i^i w))
13	12	ne0f 2258	. . . . . 6 |- (({y | ph} i^i w) =/= (/) <-> E.y y e. ({y | ph} i^i w))
14		df-rex 1626	. . . . . 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 1643	. . 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 1027	. 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.wex 956   e. wcel 1105  {cab 1440   =/= wne 1561  A.wral 1621  E.wrex 1622   i^i cin 2017  (/)c0 2251

This theorem is referenced by:  bnd 4647

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830  ax-reg 4517  ax-inf2 4549

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 773  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-rab 1628  df-v 1787  df-sbc 1913  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-if 2333  df-pw 2373  df-sn 2383  df-pr 2384  df-tp 2386  df-op 2387  df-uni 2472  df-int 2502  df-iun 2536  df-iin 2537  df-br 2588  df-opab 2635  df-tr 2649  df-eprel 2794  df-id 2797  df-po 2804  df-so 2814  df-fr 2880  df-we 2897  df-ord 2914  df-on 2915  df-lim 2916  df-suc 2917  df-om 3095  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-fv 3161  df-rdg 3871  df-r1 4567  df-rank 4568

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