Theorem inteximm 4135

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 3846	. . 3 |- ( x e. A -> |^| A C_ x )
2		vex 2733	. . . 4 |- x e. _V
3	2	ssex 4126	. . 3 |- ( |^| A C_ x -> |^| A e. _V )
4	1, 3	syl 14	. 2 |- ( x e. A -> |^| A e. _V )
5	4	exlimiv 1591	1 |- ( E. x x e. A -> |^| A e. _V )

Syntax hints:   -> wi 4   E.wex 1485   e. wcel 2141   _Vcvv 2730   C_ wss 3121   |^|cint 3831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-int 3832

This theorem is referenced by:  intexabim  4138  iinexgm  4140  onintonm  4501  elfi2  6949  elfir  6950  fifo  6957