Theorem inteximm 3924

Description: The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)

Assertion

Ref	Expression
inteximm	|- ( E. x x e. A -> |^| A e. _V )

Distinct variable group:   x, A

Proof of Theorem inteximm

Step	Hyp	Ref	Expression
1		intss1 3651	. . 3 |- ( x e. A -> |^| A C_ x )
2		vex 2604	. . . 4 |- x e. _V
3	2	ssex 3915	. . 3 |- ( |^| A C_ x -> |^| A e. _V )
4	1, 3	syl 14	. 2 |- ( x e. A -> |^| A e. _V )
5	4	exlimiv 1529	1 |- ( E. x x e. A -> |^| A e. _V )

Syntax hints:   -> wi 4   E.wex 1421   e. wcel 1433   _Vcvv 2601   C_ wss 2973   |^|cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-int 3637

This theorem is referenced by:  intexabim  3927  iinexgm  3929  onintonm  4261