Theorem en2 7065

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 6983	. . 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 5126	. . . . 5 |- ( `' f " ran f ) = dom f
4		dff1o2 5619	. . . . . . . . 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 6662	. . . . . . . 8 |- 2o = { (/) , 1o }
7	5, 6	eqtrdi 2281	. . . . . . 7 |- ( f : A -1-1-onto-> 2o -> ran f = { (/) , 1o } )
8	7	imaeq2d 5101	. . . . . 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 2279	. . . 4 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> dom f = ( `' f " { (/) , 1o } ) )
11		f1odm 5618	. . . . 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 5627	. . . . . . 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 5615	. . . . . 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 6674	. . . . . 6 |- (/) e. 2o
18	17	a1i 9	. . . . 5 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> (/) e. 2o )
19		1lt2o 6675	. . . . . 6 |- 1o e. 2o
20	19	a1i 9	. . . . 5 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> 1o e. 2o )
21		fnimapr 5737	. . . . 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 2273	. . 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 5954	. . . . 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 5954	. . . . . 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 3769	. . . . . . 7 |- ( y = ( `' f ` 1o ) -> { ( `' f ` (/) ) , y } = { ( `' f ` (/) ) , ( `' f ` 1o ) } )
30	29	eqeq2d 2244	. . . . . 6 |- ( y = ( `' f ` 1o ) -> ( A = { ( `' f ` (/) ) , y } <-> A = { ( `' f ` (/) ) , ( `' f ` 1o ) } ) )
31	30	spcegv 2905	. . . . 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 3768	. . . . . . 7 |- ( x = ( `' f ` (/) ) -> { x , y } = { ( `' f ` (/) ) , y } )
34	33	eqeq2d 2244	. . . . . 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 2905	. . . 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 1948	1 |- ( A ~~ 2o -> E. x E. y A = { x , y } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2203   (/)c0 3508   {cpr 3690   class class class wbr 4109   `'ccnv 4748   dom cdm 4749   ran crn 4750   "cima 4752   Fun wfun 5346   Fn wfn 5347   -1-1-onto->wf1o 5351   ` cfv 5352   1oc1o 6640   2oc2o 6641   ~~ cen 6973

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-1o 6647  df-2o 6648  df-en 6976

This theorem is referenced by:  en2m  7066  en2prde  7490  upgrex  16098  upgr1een  16119