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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 EvenfinA)

Proof of Theorem evennnul
Dummy variables n x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2359 . . . . . 6 (x = A → (x = (n +c n) ↔ A = (n +c n)))
21rexbidv 2635 . . . . 5 (x = A → (n Nn x = (n +c n) ↔ n Nn A = (n +c n)))
3 neeq1 2524 . . . . 5 (x = A → (xA))
42, 3anbi12d 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)}
64, 5elab2g 2987 . . 3 (A Evenfin → (A Evenfin ↔ (n Nn A = (n +c n) A)))
76ibi 232 . 2 (A Evenfin → (n Nn A = (n +c n) A))
87simprd 449 1 (A EvenfinA)
 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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