Theorem evennnul 4509

Description: An even number is nonempty. (Contributed by SF, 22-Jan-2015.)

Assertion

Ref	Expression
evennnul	|- (A e. Evenfin -> A =/= (/))

Proof of Theorem evennnul

Dummy variables n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2359	. . . . . 6 |- (x = A -> (x = (n +o n) <-> A = (n +o n)))
2	1	rexbidv 2636	. . . . 5 |- (x = A -> (E.n e. Nn x = (n +o n) <-> E.n e. Nn A = (n +o n)))
3		neeq1 2525	. . . . 5 |- (x = A -> (x =/= (/) <-> A =/= (/)))
4	2, 3	anbi12d 691	. . . 4 |- (x = A -> ((E.n e. Nn x = (n +o n) /\ x =/= (/)) <-> (E.n e. Nn A = (n +o n) /\ A =/= (/))))
5		df-evenfin 4445	. . . 4 |- Evenfin = {x | (E.n e. Nn x = (n +o n) /\ x =/= (/))}
6	4, 5	elab2g 2988	. . 3 |- (A e. Evenfin -> (A e. Evenfin <-> (E.n e. Nn A = (n +o n) /\ A =/= (/))))
7	6	ibi 232	. 2 |- (A e. Evenfin -> (E.n e. Nn A = (n +o n) /\ A =/= (/)))
8	7	simprd 449	1 |- (A e. Evenfin -> A =/= (/))

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2517  E.wrex 2616  (/)c0 3551   Nn cnnc 4374   +o cplc 4376   Even_fin cevenfin 4437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-rex 2621  df-v 2862  df-evenfin 4445

This theorem is referenced by:  evenoddnnnul  4515  vinf  4556