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Theorem 0nel1c 4159
 Description: The empty class is not a singleton. (Contributed by SF, 22-Jan-2015.)
Assertion
Ref Expression
0nel1c ¬ 1c

Proof of Theorem 0nel1c
StepHypRef Expression
1 vex 2862 . . . 4 x V
2 snprc 3788 . . . . . 6 x V ↔ {x} = )
3 eqcom 2355 . . . . . 6 ({x} = = {x})
42, 3bitri 240 . . . . 5 x V ↔ = {x})
54con1bii 321 . . . 4 = {x} ↔ x V)
61, 5mpbir 200 . . 3 ¬ = {x}
76nex 1555 . 2 ¬ x = {x}
8 el1c 4139 . 2 ( 1cx = {x})
97, 8mtbir 290 1 ¬ 1c
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ∅c0 3550  {csn 3737  1cc1c 4134 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-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 This theorem is referenced by:  pw10  4161  vfin1cltv  4547
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