Theorem bj-0nelsngl 31314

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 7190). (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 3090	. . . . . 6 |- x e. _V
2	1	snnz 4121	. . . . 5 |- { x } =/= (/)
3	2	nesymi 2704	. . . 4 |- -. (/) = { x }
4	3	nex 1674	. . 3 |- -. E. x (/) = { x }
5		bj-elsngl 31311	. . . 4 |- ( (/) e. sngl A <-> E. x e. A (/) = { x } )
6		rexex 2889	. . . 4 |- ( E. x e. A (/) = { x } -> E. x (/) = { x } )
7	5, 6	sylbi 198	. . 3 |- ( (/) e. sngl A -> E. x (/) = { x } )
8	4, 7	mto 179	. 2 |- -. (/) e. sngl A
9	8	nelir 2768	1 |- (/) e/ sngl A

Syntax hints:   = wceq 1437   E.wex 1659   e. wcel 1870   e/ wnel 2626   E.wrex 2783   (/)c0 3767   {csn 4002  sngl bj-csngl 31308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-v 3089  df-dif 3445  df-un 3447  df-nul 3768  df-sn 4003  df-pr 4005  df-bj-sngl 31309

This theorem is referenced by: (None)