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Theorem sucevenodd 4510
 Description: The successor of an even natural is odd. (Contributed by SF, 20-Jan-2015.)
Assertion
Ref Expression
sucevenodd ((A Evenfin (A +c 1c) ≠ ) → (A +c 1c) Oddfin )

Proof of Theorem sucevenodd
Dummy variables m x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2359 . . . . . . . 8 (x = A → (x = (m +c m) ↔ A = (m +c m)))
21rexbidv 2635 . . . . . . 7 (x = A → (m Nn x = (m +c m) ↔ m Nn A = (m +c m)))
3 neeq1 2524 . . . . . . 7 (x = A → (xA))
42, 3anbi12d 691 . . . . . 6 (x = A → ((m Nn x = (m +c m) x) ↔ (m Nn A = (m +c m) A)))
5 df-evenfin 4444 . . . . . 6 Evenfin = {x (m Nn x = (m +c m) x)}
64, 5elab2g 2987 . . . . 5 (A Evenfin → (A Evenfin ↔ (m Nn A = (m +c m) A)))
76ibi 232 . . . 4 (A Evenfin → (m Nn A = (m +c m) A))
8 addceq1 4383 . . . . . 6 (A = (m +c m) → (A +c 1c) = ((m +c m) +c 1c))
98reximi 2721 . . . . 5 (m Nn A = (m +c m) → m Nn (A +c 1c) = ((m +c m) +c 1c))
109adantr 451 . . . 4 ((m Nn A = (m +c m) A) → m Nn (A +c 1c) = ((m +c m) +c 1c))
117, 10syl 15 . . 3 (A Evenfinm Nn (A +c 1c) = ((m +c m) +c 1c))
1211anim1i 551 . 2 ((A Evenfin (A +c 1c) ≠ ) → (m Nn (A +c 1c) = ((m +c m) +c 1c) (A +c 1c) ≠ ))
13 1cex 4142 . . . . 5 1c V
14 addcexg 4393 . . . . 5 ((A Evenfin 1c V) → (A +c 1c) V)
1513, 14mpan2 652 . . . 4 (A Evenfin → (A +c 1c) V)
16 eqeq1 2359 . . . . . . 7 (x = (A +c 1c) → (x = ((m +c m) +c 1c) ↔ (A +c 1c) = ((m +c m) +c 1c)))
1716rexbidv 2635 . . . . . 6 (x = (A +c 1c) → (m Nn x = ((m +c m) +c 1c) ↔ m Nn (A +c 1c) = ((m +c m) +c 1c)))
18 neeq1 2524 . . . . . 6 (x = (A +c 1c) → (x ↔ (A +c 1c) ≠ ))
1917, 18anbi12d 691 . . . . 5 (x = (A +c 1c) → ((m Nn x = ((m +c m) +c 1c) x) ↔ (m Nn (A +c 1c) = ((m +c m) +c 1c) (A +c 1c) ≠ )))
20 df-oddfin 4445 . . . . 5 Oddfin = {x (m Nn x = ((m +c m) +c 1c) x)}
2119, 20elab2g 2987 . . . 4 ((A +c 1c) V → ((A +c 1c) Oddfin ↔ (m Nn (A +c 1c) = ((m +c m) +c 1c) (A +c 1c) ≠ )))
2215, 21syl 15 . . 3 (A Evenfin → ((A +c 1c) Oddfin ↔ (m Nn (A +c 1c) = ((m +c m) +c 1c) (A +c 1c) ≠ )))
2322adantr 451 . 2 ((A Evenfin (A +c 1c) ≠ ) → ((A +c 1c) Oddfin ↔ (m Nn (A +c 1c) = ((m +c m) +c 1c) (A +c 1c) ≠ )))
2412, 23mpbird 223 1 ((A Evenfin (A +c 1c) ≠ ) → (A +c 1c) Oddfin )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  Vcvv 2859  ∅c0 3550  1cc1c 4134   Nn cnnc 4373   +c cplc 4375   Evenfin cevenfin 4436   Oddfin coddfin 4437 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  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  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-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-opk 4058  df-1c 4136  df-pw1 4137  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193  df-addc 4378  df-evenfin 4444  df-oddfin 4445 This theorem is referenced by:  evenoddnnnul  4514  vinf  4555
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