Theorem en1 4513

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 4224	. . . . 5 |- 1o = {(/)}
2	1	breq2i 2677	. . . 4 |- (A ~~ 1o <-> A ~~ {(/)})
3		p0ex 2823	. . . . 5 |- {(/)} e. V
4	3	bren 4464	. . . 4 |- (A ~~ {(/)} <-> E.f f:A-1-1-onto->{(/)})
5	2, 4	bitri 171	. . 3 |- (A ~~ 1o <-> E.f f:A-1-1-onto->{(/)})
6		f1ocnv 3777	. . . . 5 |- (f:A-1-1-onto->{(/)} -> `'f:{(/)}-1-1-onto->A)
7		f1ofo 3771	. . . . . . 7 |- (`'f:{(/)}-1-1-onto->A -> `'f:{(/)}-onto->A)
8		forn 3750	. . . . . . 7 |- (`'f:{(/)}-onto->A -> ran `' f = A)
9	7, 8	syl 10	. . . . . 6 |- (`'f:{(/)}-1-1-onto->A -> ran `' f = A)
10		f1of 3765	. . . . . . . . 9 |- (`'f:{(/)}-1-1-onto->A -> `'f:{(/)}-->A)
11		0ex 2762	. . . . . . . . . . 11 |- (/) e. V
12	11	fsn2 3912	. . . . . . . . . 10 |- (`'f:{(/)}-->A <-> ((`'f` (/)) e. A /\ `'f = {<.(/), (`'f` (/))>.}))
13	12	pm3.27bi 324	. . . . . . . . 9 |- (`'f:{(/)}-->A -> `'f = {<.(/), (`'f` (/))>.})
14	10, 13	syl 10	. . . . . . . 8 |- (`'f:{(/)}-1-1-onto->A -> `'f = {<.(/), (`'f` (/))>.})
15	14	rneqd 3401	. . . . . . 7 |- (`'f:{(/)}-1-1-onto->A -> ran `' f = ran {<.(/), (`'f` (/))>.})
16		fvex 3808	. . . . . . . 8 |- (`'f` (/)) e. V
17	11, 16	rnsnop 3552	. . . . . . 7 |- ran {<.(/), (`'f` (/))>.} = {(`'f` (/))}
18	15, 17	syl6eq 1560	. . . . . 6 |- (`'f:{(/)}-1-1-onto->A -> ran `' f = {(`'f` (/))})
19	9, 18	eqtr3d 1546	. . . . 5 |- (`'f:{(/)}-1-1-onto->A -> A = {(`'f` (/))})
20		sneq 2462	. . . . . . 7 |- (x = (`'f` (/)) -> {x} = {(`'f` (/))})
21	20	eqeq2d 1523	. . . . . 6 |- (x = (`'f` (/)) -> (A = {x} <-> A = {(`'f` (/))}))
22	16, 21	cla4ev 1907	. . . . 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 1328	. . 3 |- (E.f f:A-1-1-onto->{(/)} -> E.x A = {x})
25	5, 24	sylbi 197	. 2 |- (A ~~ 1o -> E.x A = {x})
26		visset 1851	. . . . 5 |- x e. V
27	26	ensn1 4511	. . . 4 |- {x} ~~ 1o
28		breq1 2672	. . . 4 |- (A = {x} -> (A ~~ 1o <-> {x} ~~ 1o))
29	27, 28	mpbiri 192	. . 3 |- (A = {x} -> A ~~ 1o)
30	29	19.23aiv 1328	. 2 |- (E.x A = {x} -> A ~~ 1o)
31	25, 30	impbii 155	1 |- (A ~~ 1o <-> E.x A = {x})

Syntax hints:   <-> wb 144   = wceq 988   e. wcel 990  E.wex 1012  (/)c0 2324  {csn 2454  <.cop 2456   class class class wbr 2669  `'ccnv 3224  ran crn 3226  -->wf 3233  -onto->wfo 3235  -1-1-onto->wf1o 3236  ` cfv 3237  1oc1o 4212   ~~ cen 4451

This theorem is referenced by:  pm54.43 4656  card1 4920

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-nul 2761  ax-pow 2794  ax-pr 2832  ax-un 2920

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-rex 1688  df-reu 1689  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-br 2670  df-opab 2718  df-id 2889  df-suc 3009  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-f 3249  df-f1 3250  df-fo 3251  df-f1o 3252  df-fv 3253  df-1o 4217  df-en 4455

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