Theorem bj-0nelsngl 33619

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 7130). (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 3116	. . . . . 6 |- x e. _V
2	1	snnz 4145	. . . . 5 |- { x } =/= (/)
3	2	nesymi 2740	. . . 4 |- -. (/) = { x }
4	3	nex 1610	. . 3 |- -. E. x (/) = { x }
5		bj-elsngl 33616	. . . 4 |- ( (/) e. sngl A <-> E. x e. A (/) = { x } )
6		rexex 2921	. . . 4 |- ( E. x e. A (/) = { x } -> E. x (/) = { x } )
7	5, 6	sylbi 195	. . 3 |- ( (/) e. sngl A -> E. x (/) = { x } )
8	4, 7	mto 176	. 2 |- -. (/) e. sngl A
9	8	nelir 2803	1 |- (/) e/ sngl A

Syntax hints:   = wceq 1379   E.wex 1596   e. wcel 1767   e/ wnel 2663   E.wrex 2815   (/)c0 3785   {csn 4027  sngl bj-csngl 33613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-v 3115  df-dif 3479  df-un 3481  df-nul 3786  df-sn 4028  df-pr 4030  df-bj-sngl 33614

This theorem is referenced by: (None)