New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  0nelsuc Unicode version

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

Proof of Theorem 0nelsuc
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 el1c 4139 . . . . . . . . 9 1c
2 vex 2862 . . . . . . . . . . . . 13
32snid 3760 . . . . . . . . . . . 12
4 n0i 3555 . . . . . . . . . . . 12
53, 4ax-mp 5 . . . . . . . . . . 11
6 eqeq1 2359 . . . . . . . . . . 11
75, 6mtbiri 294 . . . . . . . . . 10
87exlimiv 1634 . . . . . . . . 9
91, 8sylbi 187 . . . . . . . 8 1c
10 simpr 447 . . . . . . . 8
119, 10nsyl 113 . . . . . . 7 1c
12 un00 3586 . . . . . . . . 9
13 eqcom 2355 . . . . . . . . 9
1412, 13bitri 240 . . . . . . . 8
1514notbii 287 . . . . . . 7
1611, 15sylib 188 . . . . . 6 1c
17 simpr 447 . . . . . 6
1816, 17nsyl 113 . . . . 5 1c
1918nrex 2716 . . . 4 1c
2019a1i 10 . . 3 1c
2120nrex 2716 . 2 1c
22 eladdc 4398 . 2 1c 1c
2321, 22mtbir 290 1 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   cplc 4375 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  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
 Copyright terms: Public domain W3C validator