Theorem en1bg 6624

Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.)

Assertion

Ref	Expression
en1bg	|- ( A e. V -> ( A ~~ 1o <-> A = { U. A } ) )

Proof of Theorem en1bg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		en1 6623	. . 3 |- ( A ~~ 1o <-> E. x A = { x } )
2		id 19	. . . . 5 |- ( A = { x } -> A = { x } )
3		unieq 3692	. . . . . . 7 |- ( A = { x } -> U. A = U. { x } )
4		vex 2644	. . . . . . . 8 |- x e. _V
5	4	unisn 3699	. . . . . . 7 |- U. { x } = x
6	3, 5	syl6eq 2148	. . . . . 6 |- ( A = { x } -> U. A = x )
7	6	sneqd 3487	. . . . 5 |- ( A = { x } -> { U. A } = { x } )
8	2, 7	eqtr4d 2135	. . . 4 |- ( A = { x } -> A = { U. A } )
9	8	exlimiv 1545	. . 3 |- ( E. x A = { x } -> A = { U. A } )
10	1, 9	sylbi 120	. 2 |- ( A ~~ 1o -> A = { U. A } )
11		uniexg 4299	. . . 4 |- ( A e. V -> U. A e. _V )
12		ensn1g 6621	. . . 4 |- ( U. A e. _V -> { U. A } ~~ 1o )
13	11, 12	syl 14	. . 3 |- ( A e. V -> { U. A } ~~ 1o )
14		breq1 3878	. . 3 |- ( A = { U. A } -> ( A ~~ 1o <-> { U. A } ~~ 1o ) )
15	13, 14	syl5ibrcom 156	. 2 |- ( A e. V -> ( A = { U. A } -> A ~~ 1o ) )
16	10, 15	impbid2 142	1 |- ( A e. V -> ( A ~~ 1o <-> A = { U. A } ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1299   E.wex 1436   e. wcel 1448   _Vcvv 2641   {csn 3474   U.cuni 3683   class class class wbr 3875   1oc1o 6236   ~~ cen 6562

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-reu 2382  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-suc 4231  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-1o 6243  df-en 6565

This theorem is referenced by:  en1uniel  6628