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Theorem evennnul 4508

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

Assertion

Ref	Expression
evennnul	⊢ (A ∈ Evenfin → A ≠ ∅)

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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 2635	. . . . 5 ⊢ (x = A → (∃n ∈ Nn x = (n +c n) ↔ ∃n ∈ Nn A = (n +c n)))
3		neeq1 2524	. . . . 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 4444	. . . 4 ⊢ Evenfin = {x ∣ (∃n ∈ Nn x = (n +c n) ∧ x ≠ ∅)}
6	4, 5	elab2g 2987	. . 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 ≠ ∅)

Colors of variables: wff setvar class

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  ∅c0 3550   Nn cnnc 4373   +c cplc 4375   Evenfin cevenfin 4436

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-ne 2518  df-rex 2620  df-v 2861  df-evenfin 4444

This theorem is referenced by:  evenoddnnnul  4514  vinf  4555

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