Theorem en1 4432

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 4146	. . . . 5 |- 1o = {(/)}
2	1	breq2i 2632	. . . 4 |- (A ~~ 1o <-> A ~~ {(/)})
3		p0ex 2776	. . . . 5 |- {(/)} e. V
4	3	bren 4383	. . . 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 3707	. . . . 5 |- (f:A-1-1-onto->{(/)} -> `'f:{(/)}-1-1-onto->A)
7		f1ofo 3701	. . . . . . 7 |- (`'f:{(/)}-1-1-onto->A -> `'f:{(/)}-onto->A)
8		forn 3680	. . . . . . 7 |- (`'f:{(/)}-onto->A -> ran `' f = A)
9	7, 8	syl 10	. . . . . 6 |- (`'f:{(/)}-1-1-onto->A -> ran `' f = A)
10		f1of 3695	. . . . . . . . 9 |- (`'f:{(/)}-1-1-onto->A -> `'f:{(/)}-->A)
11		0ex 2716	. . . . . . . . . . 11 |- (/) e. V
12	11	fsn2 3842	. . . . . . . . . 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 3347	. . . . . . 7 |- (`'f:{(/)}-1-1-onto->A -> ran `' f = ran {<.(/), (`'f` (/))>.})
16		fvex 3738	. . . . . . . 8 |- (`'f` (/)) e. V
17	11, 16	rnsnop 3456	. . . . . . 7 |- ran {<.(/), (`'f` (/))>.} = {(`'f` (/))}
18	15, 17	syl6eq 1526	. . . . . 6 |- (`'f:{(/)}-1-1-onto->A -> ran `' f = {(`'f` (/))})
19	9, 18	eqtr3d 1512	. . . . 5 |- (`'f:{(/)}-1-1-onto->A -> A = {(`'f` (/))})
20		sneq 2421	. . . . . . 7 |- (x = (`'f` (/)) -> {x} = {(`'f` (/))})
21	20	eqeq2d 1489	. . . . . 6 |- (x = (`'f` (/)) -> (A = {x} <-> A = {(`'f` (/))}))
22	16, 21	cla4ev 1872	. . . . 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 1297	. . 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 1816	. . . . 5 |- x e. V
27	26	ensn1 4430	. . . 4 |- {x} ~~ 1o
28		breq1 2627	. . . 4 |- (A = {x} -> (A ~~ 1o <-> {x} ~~ 1o))
29	27, 28	mpbiri 194	. . 3 |- (A = {x} -> A ~~ 1o)
30	29	19.23aiv 1297	. 2 |- (E.x A = {x} -> A ~~ 1o)
31	25, 30	impbi 157	1 |- (A ~~ 1o <-> E.x A = {x})

Syntax hints:   <-> wb 146   = wceq 958   e. wcel 960  E.wex 982  (/)c0 2283  {csn 2413  <.cop 2415   class class class wbr 2624  `'ccnv 3175  ran crn 3177  -->wf 3184  -onto->wfo 3186  -1-1-onto->wf1o 3187  ` cfv 3188  1oc1o 4134   ~~ cen 4370

This theorem is referenced by:  pm54.43 4581  card1 4843

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-reu 1654  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-suc 2960  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-1o 4139  df-en 4374

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