Theorem en2 7116

Description: A set equinumerous to ordinal 2 is an unordered pair. (Contributed by Mario Carneiro, 5-Jan-2016.)

Assertion

Ref	Expression
en2	|- ( A ~~ 2o -> E. x E. y A = { x , y } )

Distinct variable group:   x, A, y

Proof of Theorem en2

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 7034	. . 3 |- ( A ~~ 2o <-> E. f f : A -1-1-onto-> 2o )
2	1	biimpi 120	. 2 |- ( A ~~ 2o -> E. f f : A -1-1-onto-> 2o )
3		cnvimarndm 5154	. . . . 5 |- ( `' f " ran f ) = dom f
4		dff1o2 5647	. . . . . . . . 9 |- ( f : A -1-1-onto-> 2o <-> ( f Fn A /\ Fun `' f /\ ran f = 2o ) )
5	4	simp3bi 1045	. . . . . . . 8 |- ( f : A -1-1-onto-> 2o -> ran f = 2o )
6		df2o3 6706	. . . . . . . 8 |- 2o = { (/) , 1o }
7	5, 6	eqtrdi 2287	. . . . . . 7 |- ( f : A -1-1-onto-> 2o -> ran f = { (/) , 1o } )
8	7	imaeq2d 5129	. . . . . 6 |- ( f : A -1-1-onto-> 2o -> ( `' f " ran f ) = ( `' f " { (/) , 1o } ) )
9	8	adantl 277	. . . . 5 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> ( `' f " ran f ) = ( `' f " { (/) , 1o } ) )
10	3, 9	eqtr3id 2285	. . . 4 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> dom f = ( `' f " { (/) , 1o } ) )
11		f1odm 5646	. . . . 5 |- ( f : A -1-1-onto-> 2o -> dom f = A )
12	11	adantl 277	. . . 4 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> dom f = A )
13		f1ocnv 5655	. . . . . . 7 |- ( f : A -1-1-onto-> 2o -> `' f : 2o -1-1-onto-> A )
14	13	adantl 277	. . . . . 6 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> `' f : 2o -1-1-onto-> A )
15		f1ofn 5643	. . . . . 6 |- ( `' f : 2o -1-1-onto-> A -> `' f Fn 2o )
16	14, 15	syl 14	. . . . 5 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> `' f Fn 2o )
17		0lt2o 6718	. . . . . 6 |- (/) e. 2o
18	17	a1i 9	. . . . 5 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> (/) e. 2o )
19		1lt2o 6719	. . . . . 6 |- 1o e. 2o
20	19	a1i 9	. . . . 5 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> 1o e. 2o )
21		fnimapr 5766	. . . . 5 |- ( ( `' f Fn 2o /\ (/) e. 2o /\ 1o e. 2o ) -> ( `' f " { (/) , 1o } ) = { ( `' f ` (/) ) , ( `' f ` 1o ) } )
22	16, 18, 20, 21	syl3anc 1278	. . . 4 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> ( `' f " { (/) , 1o } ) = { ( `' f ` (/) ) , ( `' f ` 1o ) } )
23	10, 12, 22	3eqtr3d 2279	. . 3 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> A = { ( `' f ` (/) ) , ( `' f ` 1o ) } )
24		simpr 110	. . . . 5 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> f : A -1-1-onto-> 2o )
25		f1ocnvdm 5990	. . . . 5 |- ( ( f : A -1-1-onto-> 2o /\ (/) e. 2o ) -> ( `' f ` (/) ) e. A )
26	24, 17, 25	sylancl 417	. . . 4 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> ( `' f ` (/) ) e. A )
27		f1ocnvdm 5990	. . . . . 6 |- ( ( f : A -1-1-onto-> 2o /\ 1o e. 2o ) -> ( `' f ` 1o ) e. A )
28	24, 19, 27	sylancl 417	. . . . 5 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> ( `' f ` 1o ) e. A )
29		preq2 3789	. . . . . . 7 |- ( y = ( `' f ` 1o ) -> { ( `' f ` (/) ) , y } = { ( `' f ` (/) ) , ( `' f ` 1o ) } )
30	29	eqeq2d 2250	. . . . . 6 |- ( y = ( `' f ` 1o ) -> ( A = { ( `' f ` (/) ) , y } <-> A = { ( `' f ` (/) ) , ( `' f ` 1o ) } ) )
31	30	spcegv 2913	. . . . 5 |- ( ( `' f ` 1o ) e. A -> ( A = { ( `' f ` (/) ) , ( `' f ` 1o ) } -> E. y A = { ( `' f ` (/) ) , y } ) )
32	28, 31	syl 14	. . . 4 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> ( A = { ( `' f ` (/) ) , ( `' f ` 1o ) } -> E. y A = { ( `' f ` (/) ) , y } ) )
33		preq1 3788	. . . . . . 7 |- ( x = ( `' f ` (/) ) -> { x , y } = { ( `' f ` (/) ) , y } )
34	33	eqeq2d 2250	. . . . . 6 |- ( x = ( `' f ` (/) ) -> ( A = { x , y } <-> A = { ( `' f ` (/) ) , y } ) )
35	34	exbidv 1878	. . . . 5 |- ( x = ( `' f ` (/) ) -> ( E. y A = { x , y } <-> E. y A = { ( `' f ` (/) ) , y } ) )
36	35	spcegv 2913	. . . 4 |- ( ( `' f ` (/) ) e. A -> ( E. y A = { ( `' f ` (/) ) , y } -> E. x E. y A = { x , y } ) )
37	26, 32, 36	sylsyld 58	. . 3 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> ( A = { ( `' f ` (/) ) , ( `' f ` 1o ) } -> E. x E. y A = { x , y } ) )
38	23, 37	mpd 13	. 2 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> E. x E. y A = { x , y } )
39	2, 38	exlimddv 1954	1 |- ( A ~~ 2o -> E. x E. y A = { x , y } )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   (/)c0 3520   {cpr 3710   class class class wbr 4131   `'ccnv 4775   dom cdm 4776   ran crn 4777   "cima 4779   Fun wfun 5374   Fn wfn 5375   -1-1-onto->wf1o 5379   ` cfv 5380   1oc1o 6684   2oc2o 6685   ~~ cen 7024

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-1o 6691  df-2o 6692  df-en 7027

This theorem is used by:  en2m  7117  en2prde  7545  upgrex  16549  upgr1een  16570