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Theorem 0nelsuc 4400
 Description: The empty class is not a member of a successor. (Contributed by SF, 14-Jan-2015.)
Assertion
Ref Expression
0nelsuc ¬ (A +c 1c)

Proof of Theorem 0nelsuc
Dummy variables n m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 el1c 4139 . . . . . . . . 9 (m 1cn m = {n})
2 vex 2862 . . . . . . . . . . . . 13 n V
32snid 3760 . . . . . . . . . . . 12 n {n}
4 n0i 3555 . . . . . . . . . . . 12 (n {n} → ¬ {n} = )
53, 4ax-mp 8 . . . . . . . . . . 11 ¬ {n} =
6 eqeq1 2359 . . . . . . . . . . 11 (m = {n} → (m = ↔ {n} = ))
75, 6mtbiri 294 . . . . . . . . . 10 (m = {n} → ¬ m = )
87exlimiv 1634 . . . . . . . . 9 (n m = {n} → ¬ m = )
91, 8sylbi 187 . . . . . . . 8 (m 1c → ¬ m = )
10 simpr 447 . . . . . . . 8 ((n = m = ) → m = )
119, 10nsyl 113 . . . . . . 7 (m 1c → ¬ (n = m = ))
12 un00 3586 . . . . . . . . 9 ((n = m = ) ↔ (nm) = )
13 eqcom 2355 . . . . . . . . 9 ((nm) = = (nm))
1412, 13bitri 240 . . . . . . . 8 ((n = m = ) ↔ = (nm))
1514notbii 287 . . . . . . 7 (¬ (n = m = ) ↔ ¬ = (nm))
1611, 15sylib 188 . . . . . 6 (m 1c → ¬ = (nm))
17 simpr 447 . . . . . 6 (((nm) = = (nm)) → = (nm))
1816, 17nsyl 113 . . . . 5 (m 1c → ¬ ((nm) = = (nm)))
1918nrex 2716 . . . 4 ¬ m 1c ((nm) = = (nm))
2019a1i 10 . . 3 (n A → ¬ m 1c ((nm) = = (nm)))
2120nrex 2716 . 2 ¬ n A m 1c ((nm) = = (nm))
22 eladdc 4398 . 2 ( (A +c 1c) ↔ n A m 1c ((nm) = = (nm)))
2321, 22mtbir 290 1 ¬ (A +c 1c)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737  1cc1c 4134   +c cplc 4375 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-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ss 3259  df-nul 3551  df-sn 3741  df-1c 4136  df-addc 4378 This theorem is referenced by:  0cnsuc  4401  nndisjeq  4429  sfinltfin  4535
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