Theorem en1 4407

Description: A set is equinumerous to ordinal one iff it is a singleton.

Assertion

Ref	Expression
en1	|- (A ~~ 1o <-> E.x A = {x})

Distinct variable group:   x,A

Proof of Theorem en1

Step	Hyp	Ref	Expression
1		df1o2 4124	. . . . 5 |- 1o = {(/)}
2	1	breq2i 2617	. . . 4 |- (A ~~ 1o <-> A ~~ {(/)})
3		p0ex 2760	. . . . 5 |- {(/)} e. V
4	3	bren 4359	. . . 4 |- (A ~~ {(/)} <-> E.f f:A-1-1-onto->{(/)})
5	2, 4	bitr 173	. . 3 |- (A ~~ 1o <-> E.f f:A-1-1-onto->{(/)})
6		f1ocnv 3686	. . . . 5 |- (f:A-1-1-onto->{(/)} -> `'f:{(/)}-1-1-onto->A)
7		f1ofo 3680	. . . . . . 7 |- (`'f:{(/)}-1-1-onto->A -> `'f:{(/)}-onto->A)
8		forn 3659	. . . . . . 7 |- (`'f:{(/)}-onto->A -> ran `' f = A)
9	7, 8	syl 10	. . . . . 6 |- (`'f:{(/)}-1-1-onto->A -> ran `' f = A)
10		f1of 3674	. . . . . . . . 9 |- (`'f:{(/)}-1-1-onto->A -> `'f:{(/)}-->A)
11		0ex 2701	. . . . . . . . . . 11 |- (/) e. V
12	11	fsn2 3821	. . . . . . . . . 10 |- (`'f:{(/)}-->A <-> ((`'f` (/)) e. A /\ `'f = {<.(/), (`'f` (/))>.}))
13	12	pm3.27bi 326	. . . . . . . . 9 |- (`'f:{(/)}-->A -> `'f = {<.(/), (`'f` (/))>.})
14	10, 13	syl 10	. . . . . . . 8 |- (`'f:{(/)}-1-1-onto->A -> `'f = {<.(/), (`'f` (/))>.})
15	14	rneqd 3330	. . . . . . 7 |- (`'f:{(/)}-1-1-onto->A -> ran `' f = ran {<.(/), (`'f` (/))>.})
16		fvex 3717	. . . . . . . 8 |- (`'f` (/)) e. V
17	11, 16	rnsnop 3436	. . . . . . 7 |- ran {<.(/), (`'f` (/))>.} = {(`'f` (/))}
18	15, 17	syl6eq 1515	. . . . . 6 |- (`'f:{(/)}-1-1-onto->A -> ran `' f = {(`'f` (/))})
19	9, 18	eqtr3d 1501	. . . . 5 |- (`'f:{(/)}-1-1-onto->A -> A = {(`'f` (/))})
20		sneq 2407	. . . . . . 7 |- (x = (`'f` (/)) -> {x} = {(`'f` (/))})
21	20	eqeq2d 1478	. . . . . 6 |- (x = (`'f` (/)) -> (A = {x} <-> A = {(`'f` (/))}))
22	16, 21	cla4ev 1860	. . . . 5 |- (A = {(`'f` (/))} -> E.x A = {x})
23	6, 19, 22	3syl 20	. . . 4 |- (f:A-1-1-onto->{(/)} -> E.x A = {x})
24	23	19.23aiv 1290	. . 3 |- (E.f f:A-1-1-onto->{(/)} -> E.x A = {x})
25	5, 24	sylbi 199	. 2 |- (A ~~ 1o -> E.x A = {x})
26		visset 1804	. . . . 5 |- x e. V
27	26	ensn1 4405	. . . 4 |- {x} ~~ 1o
28		breq1 2612	. . . 4 |- (A = {x} -> (A ~~ 1o <-> {x} ~~ 1o))
29	27, 28	mpbiri 194	. . 3 |- (A = {x} -> A ~~ 1o)
30	29	19.23aiv 1290	. 2 |- (E.x A = {x} -> A ~~ 1o)
31	25, 30	impbi 157	1 |- (A ~~ 1o <-> E.x A = {x})

Syntax hints:   <-> wb 146   = wceq 953   e. wcel 955  E.wex 977  (/)c0 2270  {csn 2399  <.cop 2401   class class class wbr 2609  `'ccnv 3159  ran crn 3161  -->wf 3168  -onto->wfo 3170  -1-1-onto->wf1o 3171  ` cfv 3172  1oc1o 4112   ~~ cen 4348

This theorem is referenced by:  pm54.43 4546  card1 4805

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-reu 1643  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-1o 4117  df-en 4351

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