Theorem en1 4426

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 4140	. . . . 5 |- 1o = {(/)}
2	1	breq2i 2627	. . . 4 |- (A ~~ 1o <-> A ~~ {(/)})
3		p0ex 2770	. . . . 5 |- {(/)} e. V
4	3	bren 4377	. . . 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 3701	. . . . 5 |- (f:A-1-1-onto->{(/)} -> `'f:{(/)}-1-1-onto->A)
7		f1ofo 3695	. . . . . . 7 |- (`'f:{(/)}-1-1-onto->A -> `'f:{(/)}-onto->A)
8		forn 3674	. . . . . . 7 |- (`'f:{(/)}-onto->A -> ran `' f = A)
9	7, 8	syl 10	. . . . . 6 |- (`'f:{(/)}-1-1-onto->A -> ran `' f = A)
10		f1of 3689	. . . . . . . . 9 |- (`'f:{(/)}-1-1-onto->A -> `'f:{(/)}-->A)
11		0ex 2711	. . . . . . . . . . 11 |- (/) e. V
12	11	fsn2 3836	. . . . . . . . . 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 3341	. . . . . . 7 |- (`'f:{(/)}-1-1-onto->A -> ran `' f = ran {<.(/), (`'f` (/))>.})
16		fvex 3732	. . . . . . . 8 |- (`'f` (/)) e. V
17	11, 16	rnsnop 3450	. . . . . . 7 |- ran {<.(/), (`'f` (/))>.} = {(`'f` (/))}
18	15, 17	syl6eq 1523	. . . . . 6 |- (`'f:{(/)}-1-1-onto->A -> ran `' f = {(`'f` (/))})
19	9, 18	eqtr3d 1509	. . . . 5 |- (`'f:{(/)}-1-1-onto->A -> A = {(`'f` (/))})
20		sneq 2417	. . . . . . 7 |- (x = (`'f` (/)) -> {x} = {(`'f` (/))})
21	20	eqeq2d 1486	. . . . . 6 |- (x = (`'f` (/)) -> (A = {x} <-> A = {(`'f` (/))}))
22	16, 21	cla4ev 1869	. . . . 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 1295	. . 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 1813	. . . . 5 |- x e. V
27	26	ensn1 4424	. . . 4 |- {x} ~~ 1o
28		breq1 2622	. . . 4 |- (A = {x} -> (A ~~ 1o <-> {x} ~~ 1o))
29	27, 28	mpbiri 194	. . 3 |- (A = {x} -> A ~~ 1o)
30	29	19.23aiv 1295	. 2 |- (E.x A = {x} -> A ~~ 1o)
31	25, 30	impbi 157	1 |- (A ~~ 1o <-> E.x A = {x})

Syntax hints:   <-> wb 146   = wceq 956   e. wcel 958  E.wex 980  (/)c0 2280  {csn 2409  <.cop 2411   class class class wbr 2619  `'ccnv 3169  ran crn 3171  -->wf 3178  -onto->wfo 3180  -1-1-onto->wf1o 3181  ` cfv 3182  1oc1o 4128   ~~ cen 4364

This theorem is referenced by:  pm54.43 4572  card1 4833

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-reu 1651  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-1o 4133  df-en 4368

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