Theorem bj-0nelsngl 32959

Description: The empty set is not a member of a singletonization (neither is any nonsingleton, in particular any von Neuman ordinal except possibly df-1o 7560). (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-0nelsngl	|- (/) e/ sngl A

Proof of Theorem bj-0nelsngl

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . 6 |- x e. _V
2	1	snnz 4309	. . . . 5 |- { x } =/= (/)
3	2	nesymi 2851	. . . 4 |- -. (/) = { x }
4	3	nex 1731	. . 3 |- -. E. x (/) = { x }
5		bj-elsngl 32956	. . . 4 |- ( (/) e. sngl A <-> E. x e. A (/) = { x } )
6		rexex 3002	. . . 4 |- ( E. x e. A (/) = { x } -> E. x (/) = { x } )
7	5, 6	sylbi 207	. . 3 |- ( (/) e. sngl A -> E. x (/) = { x } )
8	4, 7	mto 188	. 2 |- -. (/) e. sngl A
9	8	nelir 2900	1 |- (/) e/ sngl A

Syntax hints:   = wceq 1483   E.wex 1704   e. wcel 1990   e/ wnel 2897   E.wrex 2913   (/)c0 3915   {csn 4177  sngl bj-csngl 32953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-bj-sngl 32954

This theorem is referenced by: (None)