Theorem en1 3331

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(s):   x,A

Proof of Theorem en1

Step	Hyp	Ref	Expression
1		df1o2 3111	. . . . 5 |- 1o = {(/)}
2	1	breq2i 2069	. . . 4 |- (A ~~ 1o <-> A ~~ {(/)})
3		p0ex 1885	. . . . 5 |- {(/)} e. V
4	3	bren 3282	. . . 4 |- (A ~~ {(/)} <-> E.f f:A-1-1-onto->{(/)})
5	2, 4	bitr 151	. . 3 |- (A ~~ 1o <-> E.f f:A-1-1-onto->{(/)})
6		f1ocnv 2811	. . . . 5 |- (f:A-1-1-onto->{(/)} -> `'f:{(/)}-1-1-onto->A)
7		f1ofo 2806	. . . . . . 7 |- (`'f:{(/)}-1-1-onto->A -> `'f:{(/)}-onto->A)
8		forn 2789	. . . . . . 7 |- (`'f:{(/)}-onto->A -> ran `'f = A)
9	7, 8	syl 12	. . . . . 6 |- (`'f:{(/)}-1-1-onto->A -> ran `'f = A)
10		f1of 2800	. . . . . . . . 9 |- (`'f:{(/)}-1-1-onto->A -> `'f:{(/)}-->A)
11		0ex 1745	. . . . . . . . . . 11 |- (/) e. V
12	11	fsn2 2896	. . . . . . . . . 10 |- (`'f:{(/)}-->A <-> ((`'f` (/)) e. A /\ `'f = {<.(/), (`'f` (/))>.}))
13	12	pm3.27bd 263	. . . . . . . . 9 |- (`'f:{(/)}-->A -> `'f = {<.(/), (`'f` (/))>.})
14	10, 13	syl 12	. . . . . . . 8 |- (`'f:{(/)}-1-1-onto->A -> `'f = {<.(/), (`'f` (/))>.})
15	14	rneqd 2557	. . . . . . 7 |- (`'f:{(/)}-1-1-onto->A -> ran `'f = ran {<.(/), (`'f` (/))>.})
16		fvex 2838	. . . . . . . 8 |- (`'f` (/)) e. V
17	11, 16	rnsnop 2637	. . . . . . 7 |- ran {<.(/), (`'f` (/))>.} = {(`'f` (/))}
18	15, 17	syl6eq 1140	. . . . . 6 |- (`'f:{(/)}-1-1-onto->A -> ran `'f = {(`'f` (/))})
19	9, 18	eqtr3d 1130	. . . . 5 |- (`'f:{(/)}-1-1-onto->A -> A = {(`'f` (/))})
20		sneq 1816	. . . . . . 7 |- (x = (`'f` (/)) -> {x} = {(`'f` (/))})
21	20	cleq2d 1112	. . . . . 6 |- (x = (`'f` (/)) -> (A = {x} <-> A = {(`'f` (/))}))
22	16, 21	cla4ev 1401	. . . . 5 |- (A = {(`'f` (/))} -> E.x A = {x})
23	6, 19, 22	3syl 21	. . . 4 |- (f:A-1-1-onto->{(/)} -> E.x A = {x})
24	23	19.23aiv 952	. . 3 |- (E.f f:A-1-1-onto->{(/)} -> E.x A = {x})
25	5, 24	sylbi 174	. 2 |- (A ~~ 1o -> E.x A = {x})
26		visset 1350	. . . . 5 |- x e. V
27	26	ensn1 3329	. . . 4 |- {x} ~~ 1o
28		breq1 2065	. . . 4 |- (A = {x} -> (A ~~ 1o <-> {x} ~~ 1o))
29	27, 28	mpbiri 169	. . 3 |- (A = {x} -> A ~~ 1o)
30	29	19.23aiv 952	. 2 |- (E.x A = {x} -> A ~~ 1o)
31	25, 30	impbi 139	1 |- (A ~~ 1o <-> E.x A = {x})

Syntax hints:   <-> wb 127  E.wex 678   = wceq 1091   e. wcel 1092  (/)c0 1707  {csn 1808  <.cop 1810   class class class wbr 2054  `'ccnv 2409  ran crn 2411  -->wf 2418  -onto->wfo 2420  -1-1-onto->wf1o 2421  ` cfv 2422  1oc1o 3099   ~~ cen 3271

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-reu 1207  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-1o 3104  df-en 3274

metamath.org