Theorem en1bg 7080

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 7079	. . 3 |- ( A ~~ 1o <-> E. x A = { x } )
2		id 19	. . . . 5 |- ( A = { x } -> A = { x } )
3		unieq 3942	. . . . . . 7 |- ( A = { x } -> U. A = U. { x } )
4		vex 2824	. . . . . . . 8 |- x e. _V
5	4	unisn 3949	. . . . . . 7 |- U. { x } = x
6	3, 5	eqtrdi 2287	. . . . . 6 |- ( A = { x } -> U. A = x )
7	6	sneqd 3721	. . . . 5 |- ( A = { x } -> { U. A } = { x } )
8	2, 7	eqtr4d 2274	. . . 4 |- ( A = { x } -> A = { U. A } )
9	8	exlimiv 1651	. . 3 |- ( E. x A = { x } -> A = { U. A } )
10	1, 9	sylbi 121	. 2 |- ( A ~~ 1o -> A = { U. A } )
11		uniexg 4583	. . . 4 |- ( A e. V -> U. A e. _V )
12		ensn1g 7077	. . . 4 |- ( U. A e. _V -> { U. A } ~~ 1o )
13	11, 12	syl 14	. . 3 |- ( A e. V -> { U. A } ~~ 1o )
14		breq1 4131	. . 3 |- ( A = { U. A } -> ( A ~~ 1o <-> { U. A } ~~ 1o ) )
15	13, 14	syl5ibrcom 157	. 2 |- ( A e. V -> ( A = { U. A } -> A ~~ 1o ) )
16	10, 15	impbid2 143	1 |- ( A e. V -> ( A ~~ 1o <-> A = { U. A } ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   {csn 3708   U.cuni 3933   class class class wbr 4128   1oc1o 6673   ~~ cen 7013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-suc 4514  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-1o 6680  df-en 7016

This theorem is referenced by:  en1uniel  7084