Theorem stcat 10457

Description: Structure of the class abstraction used by Alg, Cat and Ded.

Assertion

Ref	Expression
stcat	|- {x | E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph)} (_ ((V X. V) X. (V X. V))

Distinct variable groups:   x,v   x,w   x,y   x,z

Proof of Theorem stcat

Step	Hyp	Ref	Expression
1		opex 2782	. . . . . . . 8 |- <.v, w>. e. V
2	1	opelxp 3214	. . . . . . 7 |- (<.<.y, z>., <.v, w>.>. e. ((V X. V) X. (V X. V)) <-> (<.y, z>. e. (V X. V) /\ <.v, w>. e. (V X. V)))
3		visset 1813	. . . . . . . . 9 |- z e. V
4	3	opelxp 3214	. . . . . . . 8 |- (<.y, z>. e. (V X. V) <-> (y e. V /\ z e. V))
5		visset 1813	. . . . . . . 8 |- y e. V
6	4, 5, 3	mpbir2an 730	. . . . . . 7 |- <.y, z>. e. (V X. V)
7		visset 1813	. . . . . . . . 9 |- w e. V
8	7	opelxp 3214	. . . . . . . 8 |- (<.v, w>. e. (V X. V) <-> (v e. V /\ w e. V))
9		visset 1813	. . . . . . . 8 |- v e. V
10	8, 9, 7	mpbir2an 730	. . . . . . 7 |- <.v, w>. e. (V X. V)
11	2, 6, 10	mpbir2an 730	. . . . . 6 |- <.<.y, z>., <.v, w>.>. e. ((V X. V) X. (V X. V))
12		eleq1 1534	. . . . . 6 |- (x = <.<.y, z>., <.v, w>.>. -> (x e. ((V X. V) X. (V X. V)) <-> <.<.y, z>., <.v, w>.>. e. ((V X. V) X. (V X. V))))
13	11, 12	mpbiri 194	. . . . 5 |- (x = <.<.y, z>., <.v, w>.>. -> x e. ((V X. V) X. (V X. V)))
14	13	adantr 389	. . . 4 |- ((x = <.<.y, z>., <.v, w>.>. /\ ph) -> x e. ((V X. V) X. (V X. V)))
15	14	19.23aivv 1296	. . 3 |- (E.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph) -> x e. ((V X. V) X. (V X. V)))
16	15	19.23aivv 1296	. 2 |- (E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph) -> x e. ((V X. V) X. (V X. V)))
17	16	abssi 2122	1 |- {x | E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph)} (_ ((V X. V) X. (V X. V))

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  Vcvv 1811   (_ wss 2047  <.cop 2411   X. cxp 3168

This theorem is referenced by:  strded 10672  strcat 10693

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-opab 2667  df-xp 3184

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