Theorem bj-snglex 32961

Description: A class is a set if and only if its singletonization is a set. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-snglex	|- ( A e. _V <-> sngl A e. _V )

Proof of Theorem bj-snglex

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isset 3207	. . 3 |- ( A e. _V <-> E. x x = A )
2		pweq 4161	. . . . 5 |- ( x = A -> ~P x = ~P A )
3	2	eximi 1762	. . . 4 |- ( E. x x = A -> E. x ~P x = ~P A )
4		bj-snglss 32958	. . . . . 6 |- sngl A C_ ~P A
5		sseq2 3627	. . . . . 6 |- ( ~P x = ~P A -> (sngl A C_ ~P x <-> sngl A C_ ~P A ) )
6	4, 5	mpbiri 248	. . . . 5 |- ( ~P x = ~P A -> sngl A C_ ~P x )
7	6	eximi 1762	. . . 4 |- ( E. x ~P x = ~P A -> E. xsngl A C_ ~P x )
8		vpwex 4849	. . . . . 6 |- ~P x e. _V
9	8	ssex 4802	. . . . 5 |- (sngl A C_ ~P x -> sngl A e. _V )
10	9	exlimiv 1858	. . . 4 |- ( E. xsngl A C_ ~P x -> sngl A e. _V )
11	3, 7, 10	3syl 18	. . 3 |- ( E. x x = A -> sngl A e. _V )
12	1, 11	sylbi 207	. 2 |- ( A e. _V -> sngl A e. _V )
13		bj-snglinv 32960	. . 3 |- A = { y | { y } e. sngl A }
14		bj-snsetex 32951	. . 3 |- (sngl A e. _V -> { y | { y } e. sngl A } e. _V )
15	13, 14	syl5eqel 2705	. 2 |- (sngl A e. _V -> A e. _V )
16	12, 15	impbii 199	1 |- ( A e. _V <-> sngl A e. _V )

Syntax hints:   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177  sngl bj-csngl 32953

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-csb 3534  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-bj-sngl 32954

This theorem is referenced by:  bj-tagex  32975