Theorem bj-snmoore 33068

Description: A singleton is a Moore collection. (Contributed by BJ, 9-Dec-2021.)

Assertion

Ref	Expression
bj-snmoore	|- ( A e. _V <-> { A } e. Moore_ )

Proof of Theorem bj-snmoore

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4908	. . . 4 |- { A } e. _V
2	1	a1i 11	. . 3 |- ( A e. _V -> { A } e. _V )
3		unisng 4452	. . . 4 |- ( A e. _V -> U. { A } = A )
4		snidg 4206	. . . 4 |- ( A e. _V -> A e. { A } )
5	3, 4	eqeltrd 2701	. . 3 |- ( A e. _V -> U. { A } e. { A } )
6		df-ne 2795	. . . . . . . 8 |- ( x =/= (/) <-> -. x = (/) )
7		sssn 4358	. . . . . . . 8 |- ( x C_ { A } <-> ( x = (/) \/ x = { A } ) )
8		biorf 420	. . . . . . . . 9 |- ( -. x = (/) -> ( x = { A } <-> ( x = (/) \/ x = { A } ) ) )
9	8	biimpar 502	. . . . . . . 8 |- ( ( -. x = (/) /\ ( x = (/) \/ x = { A } ) ) -> x = { A } )
10	6, 7, 9	syl2anb 496	. . . . . . 7 |- ( ( x =/= (/) /\ x C_ { A } ) -> x = { A } )
11		inteq 4478	. . . . . . . . 9 |- ( x = { A } -> |^| x = |^| { A } )
12		intsng 4512	. . . . . . . . 9 |- ( A e. _V -> |^| { A } = A )
13		eqtr 2641	. . . . . . . . . 10 |- ( ( |^| x = |^| { A } /\ |^| { A } = A ) -> |^| x = A )
14	13	ex 450	. . . . . . . . 9 |- ( |^| x = |^| { A } -> ( |^| { A } = A -> |^| x = A ) )
15	11, 12, 14	syl2im 40	. . . . . . . 8 |- ( x = { A } -> ( A e. _V -> |^| x = A ) )
16		intex 4820	. . . . . . . . . 10 |- ( x =/= (/) <-> |^| x e. _V )
17		elsng 4191	. . . . . . . . . 10 |- ( |^| x e. _V -> ( |^| x e. { A } <-> |^| x = A ) )
18	16, 17	sylbi 207	. . . . . . . . 9 |- ( x =/= (/) -> ( |^| x e. { A } <-> |^| x = A ) )
19	18	biimprd 238	. . . . . . . 8 |- ( x =/= (/) -> ( |^| x = A -> |^| x e. { A } ) )
20	15, 19	sylan9r 690	. . . . . . 7 |- ( ( x =/= (/) /\ x = { A } ) -> ( A e. _V -> |^| x e. { A } ) )
21	10, 20	syldan 487	. . . . . 6 |- ( ( x =/= (/) /\ x C_ { A } ) -> ( A e. _V -> |^| x e. { A } ) )
22	21	ex 450	. . . . 5 |- ( x =/= (/) -> ( x C_ { A } -> ( A e. _V -> |^| x e. { A } ) ) )
23	22	com13 88	. . . 4 |- ( A e. _V -> ( x C_ { A } -> ( x =/= (/) -> |^| x e. { A } ) ) )
24	23	imp31 448	. . 3 |- ( ( ( A e. _V /\ x C_ { A } ) /\ x =/= (/) ) -> |^| x e. { A } )
25	2, 5, 24	bj-ismooredr2 33065	. 2 |- ( A e. _V -> { A } e. Moore_ )
26		snprc 4253	. . . . 5 |- ( -. A e. _V <-> { A } = (/) )
27	26	biimpi 206	. . . 4 |- ( -. A e. _V -> { A } = (/) )
28		bj-0nmoore 33067	. . . . 5 |- -. (/) e. Moore_
29	28	a1i 11	. . . 4 |- ( -. A e. _V -> -. (/) e. Moore_ )
30	27, 29	eqneltrd 2720	. . 3 |- ( -. A e. _V -> -. { A } e. Moore_ )
31	30	con4i 113	. 2 |- ( { A } e. Moore_ -> A e. _V )
32	25, 31	impbii 199	1 |- ( A e. _V <-> { A } e. Moore_ )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436   |^|cint 4475  Moore_cmoore 33057

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-8 1992  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-3an 1039  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-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-int 4476  df-bj-moore 33058

This theorem is referenced by: (None)