Theorem bj-0nelsngl 31635

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 7200). (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 3034	. . . . . 6 |- x e. _V
2	1	snnz 4081	. . . . 5 |- { x } =/= (/)
3	2	nesymi 2700	. . . 4 |- -. (/) = { x }
4	3	nex 1686	. . 3 |- -. E. x (/) = { x }
5		bj-elsngl 31632	. . . 4 |- ( (/) e. sngl A <-> E. x e. A (/) = { x } )
6		rexex 2843	. . . 4 |- ( E. x e. A (/) = { x } -> E. x (/) = { x } )
7	5, 6	sylbi 200	. . 3 |- ( (/) e. sngl A -> E. x (/) = { x } )
8	4, 7	mto 181	. 2 |- -. (/) e. sngl A
9	8	nelir 2746	1 |- (/) e/ sngl A

Syntax hints:   = wceq 1452   E.wex 1671   e. wcel 1904   e/ wnel 2642   E.wrex 2757   (/)c0 3722   {csn 3959  sngl bj-csngl 31629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-un 3395  df-nul 3723  df-sn 3960  df-pr 3962  df-bj-sngl 31630

This theorem is referenced by: (None)