Theorem 0nel1c 4160

Description: The empty class is not a singleton. (Contributed by SF, 22-Jan-2015.)

Assertion

Ref	Expression
0nel1c	⊢ ¬ ∅ ∈ 1c

Proof of Theorem 0nel1c

Step	Hyp	Ref	Expression
1		vex 2863	. . . 4 ⊢ x ∈ V
2		snprc 3789	. . . . . 6 ⊢ (¬ x ∈ V ↔ {x} = ∅)
3		eqcom 2355	. . . . . 6 ⊢ ({x} = ∅ ↔ ∅ = {x})
4	2, 3	bitri 240	. . . . 5 ⊢ (¬ x ∈ V ↔ ∅ = {x})
5	4	con1bii 321	. . . 4 ⊢ (¬ ∅ = {x} ↔ x ∈ V)
6	1, 5	mpbir 200	. . 3 ⊢ ¬ ∅ = {x}
7	6	nex 1555	. 2 ⊢ ¬ ∃x∅ = {x}
8		el1c 4140	. 2 ⊢ (∅ ∈ 1c ↔ ∃x∅ = {x})
9	7, 8	mtbir 290	1 ⊢ ¬ ∅ ∈ 1c

Syntax hints:  ¬ wn 3  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ∅c0 3551  {csn 3738  1_cc1c 4135

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 4079  ax-sn 4088

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 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-1c 4137

This theorem is referenced by:  pw10  4162  vfin1cltv  4548