Theorem inteximm 4110

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 3822	. . 3 |- ( x e. A -> |^| A C_ x )
2		vex 2715	. . . 4 |- x e. _V
3	2	ssex 4101	. . 3 |- ( |^| A C_ x -> |^| A e. _V )
4	1, 3	syl 14	. 2 |- ( x e. A -> |^| A e. _V )
5	4	exlimiv 1578	1 |- ( E. x x e. A -> |^| A e. _V )

Syntax hints:   -> wi 4   E.wex 1472   e. wcel 2128   _Vcvv 2712   C_ wss 3102   |^|cint 3807

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-sep 4082

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-int 3808

This theorem is referenced by:  intexabim  4113  iinexgm  4115  onintonm  4476  elfi2  6916  elfir  6917  fifo  6924