Theorem en1 4567

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 4276	. . . . 5 |- 1o = {(/)}
2	1	breq2i 2700	. . . 4 |- (A ~~ 1o <-> A ~~ {(/)})
3		p0ex 2828	. . . . 5 |- {(/)} e. V
4	3	bren 4518	. . . 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 3809	. . . . 5 |- (f:A-1-1-onto->{(/)} -> `'f:{(/)}-1-1-onto->A)
7		f1ofo 3803	. . . . . . 7 |- (`'f:{(/)}-1-1-onto->A -> `'f:{(/)}-onto->A)
8		forn 3782	. . . . . . 7 |- (`'f:{(/)}-onto->A -> ran `' f = A)
9	7, 8	syl 10	. . . . . 6 |- (`'f:{(/)}-1-1-onto->A -> ran `' f = A)
10		f1of 3797	. . . . . . . . 9 |- (`'f:{(/)}-1-1-onto->A -> `'f:{(/)}-->A)
11		0ex 2785	. . . . . . . . . . 11 |- (/) e. V
12	11	fsn2 3950	. . . . . . . . . 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 3428	. . . . . . 7 |- (`'f:{(/)}-1-1-onto->A -> ran `' f = ran {<.(/), (`'f` (/))>.})
16		fvex 3843	. . . . . . . 8 |- (`'f` (/)) e. V
17	11, 16	rnsnop 3582	. . . . . . 7 |- ran {<.(/), (`'f` (/))>.} = {(`'f` (/))}
18	15, 17	syl6eq 1566	. . . . . 6 |- (`'f:{(/)}-1-1-onto->A -> ran `' f = {(`'f` (/))})
19	9, 18	eqtr3d 1552	. . . . 5 |- (`'f:{(/)}-1-1-onto->A -> A = {(`'f` (/))})
20		sneq 2475	. . . . . . 7 |- (x = (`'f` (/)) -> {x} = {(`'f` (/))})
21	20	eqeq2d 1529	. . . . . 6 |- (x = (`'f` (/)) -> (A = {x} <-> A = {(`'f` (/))}))
22	16, 21	cla4ev 1915	. . . . 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 1333	. . 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 1859	. . . . 5 |- x e. V
27	26	ensn1 4565	. . . 4 |- {x} ~~ 1o
28		breq1 2695	. . . 4 |- (A = {x} -> (A ~~ 1o <-> {x} ~~ 1o))
29	27, 28	mpbiri 192	. . 3 |- (A = {x} -> A ~~ 1o)
30	29	19.23aiv 1333	. 2 |- (E.x A = {x} -> A ~~ 1o)
31	25, 30	impbii 155	1 |- (A ~~ 1o <-> E.x A = {x})

Syntax hints:   <-> wb 144   = wceq 992   e. wcel 994  E.wex 1016  (/)c0 2332  {csn 2467  <.cop 2469   class class class wbr 2692  `'ccnv 3250  ran crn 3252  -->wf 3259  -onto->wfo 3261  -1-1-onto->wf1o 3262  ` cfv 3263  1oc1o 4264   ~~ cen 4505

This theorem is referenced by:  pm54.43 4715  card1 4981

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-id 2913  df-suc 2981  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-1o 4269  df-en 4509

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