Theorem evennnul 4509

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

Assertion

Ref	Expression
evennnul	⊢ (A ∈ 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 +c n) ↔ A = (n +c n)))
2	1	rexbidv 2636	. . . . 5 ⊢ (x = A → (∃n ∈ Nn x = (n +c n) ↔ ∃n ∈ Nn A = (n +c n)))
3		neeq1 2525	. . . . 5 ⊢ (x = A → (x ≠ ∅ ↔ A ≠ ∅))
4	2, 3	anbi12d 691	. . . 4 ⊢ (x = A → ((∃n ∈ Nn x = (n +c n) ∧ x ≠ ∅) ↔ (∃n ∈ Nn A = (n +c n) ∧ A ≠ ∅)))
5		df-evenfin 4445	. . . 4 ⊢ Evenfin = {x ∣ (∃n ∈ Nn x = (n +c n) ∧ x ≠ ∅)}
6	4, 5	elab2g 2988	. . 3 ⊢ (A ∈ Evenfin → (A ∈ Evenfin ↔ (∃n ∈ Nn A = (n +c n) ∧ A ≠ ∅)))
7	6	ibi 232	. 2 ⊢ (A ∈ Evenfin → (∃n ∈ Nn A = (n +c n) ∧ A ≠ ∅))
8	7	simprd 449	1 ⊢ (A ∈ Evenfin → A ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616  ∅c0 3551   Nn cnnc 4374   +_c 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