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Theorem oddnnul 4509
 Description: An odd number is nonempty. (Contributed by SF, 22-Jan-2015.)
Assertion
Ref Expression
oddnnul (A OddfinA)

Proof of Theorem oddnnul
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) +c 1c) ↔ A = ((n +c n) +c 1c)))
21rexbidv 2635 . . . . 5 (x = A → (n Nn x = ((n +c n) +c 1c) ↔ n Nn A = ((n +c n) +c 1c)))
3 neeq1 2524 . . . . 5 (x = A → (xA))
42, 3anbi12d 691 . . . 4 (x = A → ((n Nn x = ((n +c n) +c 1c) x) ↔ (n Nn A = ((n +c n) +c 1c) A)))
5 df-oddfin 4445 . . . 4 Oddfin = {x (n Nn x = ((n +c n) +c 1c) x)}
64, 5elab2g 2987 . . 3 (A Oddfin → (A Oddfin ↔ (n Nn A = ((n +c n) +c 1c) A)))
76ibi 232 . 2 (A Oddfin → (n Nn A = ((n +c n) +c 1c) A))
87simprd 449 1 (A OddfinA)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  ∅c0 3550  1cc1c 4134   Nn cnnc 4373   +c cplc 4375   Oddfin coddfin 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 2478  df-ne 2518  df-rex 2620  df-v 2861  df-oddfin 4445 This theorem is referenced by:  evenoddnnnul  4514  vinf  4555
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