Theorem inteximm 4082

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 3794	. . 3 |- ( x e. A -> |^| A C_ x )
2		vex 2692	. . . 4 |- x e. _V
3	2	ssex 4073	. . 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 1469   e. wcel 1481   _Vcvv 2689   C_ wss 3076   |^|cint 3779

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-int 3780

This theorem is referenced by:  intexabim  4085  iinexgm  4087  onintonm  4441  elfi2  6868  elfir  6869  fifo  6876