Theorem stcat 14347

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)} C_ ((_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 3527	. . . . . . . 8 |- <.v, w>. e. _V
2	1	opelxp 4036	. . . . . . 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 2295	. . . . . . . . 9 |- z e. _V
4	3	opelxp 4036	. . . . . . . 8 |- (<.y, z>. e. (_V X. _V) <-> (y e. _V /\ z e. _V))
5		visset 2295	. . . . . . . 8 |- y e. _V
6	4, 5, 3	mpbir2an 800	. . . . . . 7 |- <.y, z>. e. (_V X. _V)
7		visset 2295	. . . . . . . . 9 |- w e. _V
8	7	opelxp 4036	. . . . . . . 8 |- (<.v, w>. e. (_V X. _V) <-> (v e. _V /\ w e. _V))
9		visset 2295	. . . . . . . 8 |- v e. _V
10	8, 9, 7	mpbir2an 800	. . . . . . 7 |- <.v, w>. e. (_V X. _V)
11	2, 6, 10	mpbir2an 800	. . . . . 6 |- <.<.y, z>., <.v, w>.>. e. ((_V X. _V) X. (_V X. _V))
12		eleq1 1957	. . . . . 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 211	. . . . 5 |- (x = <.<.y, z>., <.v, w>.>. -> x e. ((_V X. _V) X. (_V X. _V)))
14	13	adantr 425	. . . 4 |- ((x = <.<.y, z>., <.v, w>.>. /\ ph) -> x e. ((_V X. _V) X. (_V X. _V)))
15	14	19.23aivv 1675	. . 3 |- (E.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph) -> x e. ((_V X. _V) X. (_V X. _V)))
16	15	19.23aivv 1675	. 2 |- (E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph) -> x e. ((_V X. _V) X. (_V X. _V)))
17	16	abssi 2682	1 |- {x | E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph)} C_ ((_V X. _V) X. (_V X. _V))

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  _Vcvv 2292   C_ wss 2593  <.cop 3046   X. cxp 3984

This theorem is referenced by:  strded 15086  strcat 15107

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-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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-opab 3396  df-xp 4000

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