Theorem inteximm 4198

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 3903	. . 3 |- ( x e. A -> |^| A C_ x )
2		vex 2776	. . . 4 |- x e. _V
3	2	ssex 4186	. . 3 |- ( |^| A C_ x -> |^| A e. _V )
4	1, 3	syl 14	. 2 |- ( x e. A -> |^| A e. _V )
5	4	exlimiv 1622	1 |- ( E. x x e. A -> |^| A e. _V )

Syntax hints:   -> wi 4   E.wex 1516   e. wcel 2177   _Vcvv 2773   C_ wss 3168   |^|cint 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-sep 4167

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3174  df-ss 3181  df-int 3889

This theorem is referenced by:  intexabim  4201  iinexgm  4203  onintonm  4570  elfi2  7086  elfir  7087  fifo  7094