Theorem cexint2 14862

Description: Conditions so that an intersection exists.

Assertion

Ref	Expression
cexint2	|- ((A =/= (/) /\ A.x e. A B e. C) -> |^|_x e. A B e. _V)

Distinct variable group:   x,A

Proof of Theorem cexint2

Step	Hyp	Ref	Expression
1		dfiin2g 3286	. . 3 |- (A.x e. A B e. C -> |^|_x e. A B = |^|{y | E.x e. A y = B})
2	1	adantl 424	. 2 |- ((A =/= (/) /\ A.x e. A B e. C) -> |^|_x e. A B = |^|{y | E.x e. A y = B})
3		elex 2302	. . . . . . . . . 10 |- (B e. C -> E.y y = B)
4	3	a1i 8	. . . . . . . . 9 |- (x e. A -> (B e. C -> E.y y = B))
5	4	rgen 2159	. . . . . . . 8 |- A.x e. A (B e. C -> E.y y = B)
6		r19.2z 2958	. . . . . . . 8 |- ((A =/= (/) /\ A.x e. A (B e. C -> E.y y = B)) -> E.x e. A (B e. C -> E.y y = B))
7	5, 6	mpan2 760	. . . . . . 7 |- (A =/= (/) -> E.x e. A (B e. C -> E.y y = B))
8		r19.35 2231	. . . . . . 7 |- (E.x e. A (B e. C -> E.y y = B) <-> (A.x e. A B e. C -> E.x e. A E.y y = B))
9	7, 8	sylib 215	. . . . . 6 |- (A =/= (/) -> (A.x e. A B e. C -> E.x e. A E.y y = B))
10	9	imp 377	. . . . 5 |- ((A =/= (/) /\ A.x e. A B e. C) -> E.x e. A E.y y = B)
11		rexcom4 2312	. . . . 5 |- (E.x e. A E.y y = B <-> E.yE.x e. A y = B)
12	10, 11	sylib 215	. . . 4 |- ((A =/= (/) /\ A.x e. A B e. C) -> E.yE.x e. A y = B)
13		abn0 2892	. . . 4 |- ({y | E.x e. A y = B} =/= (/) <-> E.yE.x e. A y = B)
14	12, 13	sylibr 217	. . 3 |- ((A =/= (/) /\ A.x e. A B e. C) -> {y | E.x e. A y = B} =/= (/))
15		intex 3465	. . 3 |- ({y | E.x e. A y = B} =/= (/) <-> |^|{y | E.x e. A y = B} e. _V)
16	14, 15	sylib 215	. 2 |- ((A =/= (/) /\ A.x e. A B e. C) -> |^|{y | E.x e. A y = B} e. _V)
17	2, 16	eqeltrd 1971	1 |- ((A =/= (/) /\ A.x e. A B e. C) -> |^|_x e. A B e. _V)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292  (/)c0 2875  |^|cint 3214  |^|_ciin 3256

This theorem is referenced by:  inttop2 14863

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-13 1311  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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-int 3215  df-iin 3258

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