Theorem en2 7078

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 6996	. . 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 5131	. . . . 5 |- ( `' f " ran f ) = dom f
4		dff1o2 5624	. . . . . . . . 9 |- ( f : A -1-1-onto-> 2o <-> ( f Fn A /\ Fun `' f /\ ran f = 2o ) )
5	4	simp3bi 1041	. . . . . . . 8 |- ( f : A -1-1-onto-> 2o -> ran f = 2o )
6		df2o3 6675	. . . . . . . 8 |- 2o = { (/) , 1o }
7	5, 6	eqtrdi 2283	. . . . . . 7 |- ( f : A -1-1-onto-> 2o -> ran f = { (/) , 1o } )
8	7	imaeq2d 5106	. . . . . 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 2281	. . . 4 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> dom f = ( `' f " { (/) , 1o } ) )
11		f1odm 5623	. . . . 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 5632	. . . . . . 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 5620	. . . . . 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 6687	. . . . . 6 |- (/) e. 2o
18	17	a1i 9	. . . . 5 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> (/) e. 2o )
19		1lt2o 6688	. . . . . 6 |- 1o e. 2o
20	19	a1i 9	. . . . 5 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> 1o e. 2o )
21		fnimapr 5742	. . . . 5 |- ( ( `' f Fn 2o /\ (/) e. 2o /\ 1o e. 2o ) -> ( `' f " { (/) , 1o } ) = { ( `' f ` (/) ) , ( `' f ` 1o ) } )
22	16, 18, 20, 21	syl3anc 1274	. . . 4 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> ( `' f " { (/) , 1o } ) = { ( `' f ` (/) ) , ( `' f ` 1o ) } )
23	10, 12, 22	3eqtr3d 2275	. . 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 5960	. . . . 5 |- ( ( f : A -1-1-onto-> 2o /\ (/) e. 2o ) -> ( `' f ` (/) ) e. A )
26	24, 17, 25	sylancl 413	. . . 4 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> ( `' f ` (/) ) e. A )
27		f1ocnvdm 5960	. . . . . 6 |- ( ( f : A -1-1-onto-> 2o /\ 1o e. 2o ) -> ( `' f ` 1o ) e. A )
28	24, 19, 27	sylancl 413	. . . . 5 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> ( `' f ` 1o ) e. A )
29		preq2 3774	. . . . . . 7 |- ( y = ( `' f ` 1o ) -> { ( `' f ` (/) ) , y } = { ( `' f ` (/) ) , ( `' f ` 1o ) } )
30	29	eqeq2d 2246	. . . . . 6 |- ( y = ( `' f ` 1o ) -> ( A = { ( `' f ` (/) ) , y } <-> A = { ( `' f ` (/) ) , ( `' f ` 1o ) } ) )
31	30	spcegv 2907	. . . . 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 3773	. . . . . . 7 |- ( x = ( `' f ` (/) ) -> { x , y } = { ( `' f ` (/) ) , y } )
34	33	eqeq2d 2246	. . . . . 6 |- ( x = ( `' f ` (/) ) -> ( A = { x , y } <-> A = { ( `' f ` (/) ) , y } ) )
35	34	exbidv 1874	. . . . 5 |- ( x = ( `' f ` (/) ) -> ( E. y A = { x , y } <-> E. y A = { ( `' f ` (/) ) , y } ) )
36	35	spcegv 2907	. . . 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 1950	1 |- ( A ~~ 2o -> E. x E. y A = { x , y } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2205   (/)c0 3512   {cpr 3695   class class class wbr 4114   `'ccnv 4753   dom cdm 4754   ran crn 4755   "cima 4757   Fun wfun 5351   Fn wfn 5352   -1-1-onto->wf1o 5356   ` cfv 5357   1oc1o 6653   2oc2o 6654   ~~ cen 6986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1o 6660  df-2o 6661  df-en 6989

This theorem is referenced by:  en2m  7079  en2prde  7503  upgrex  16224  upgr1een  16245