Theorem bj-0nelsngl 31565

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 7182). (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 3048	. . . . . 6 |- x e. _V
2	1	snnz 4090	. . . . 5 |- { x } =/= (/)
3	2	nesymi 2681	. . . 4 |- -. (/) = { x }
4	3	nex 1678	. . 3 |- -. E. x (/) = { x }
5		bj-elsngl 31562	. . . 4 |- ( (/) e. sngl A <-> E. x e. A (/) = { x } )
6		rexex 2844	. . . 4 |- ( E. x e. A (/) = { x } -> E. x (/) = { x } )
7	5, 6	sylbi 199	. . 3 |- ( (/) e. sngl A -> E. x (/) = { x } )
8	4, 7	mto 180	. 2 |- -. (/) e. sngl A
9	8	nelir 2727	1 |- (/) e/ sngl A

Syntax hints:   = wceq 1444   E.wex 1663   e. wcel 1887   e/ wnel 2623   E.wrex 2738   (/)c0 3731   {csn 3968  sngl bj-csngl 31559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-v 3047  df-dif 3407  df-un 3409  df-nul 3732  df-sn 3969  df-pr 3971  df-bj-sngl 31560

This theorem is referenced by: (None)