Theorem en2 7041

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 6960	. . 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 5107	. . . . 5 |- ( `' f " ran f ) = dom f
4		dff1o2 5597	. . . . . . . . 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 6640	. . . . . . . 8 |- 2o = { (/) , 1o }
7	5, 6	eqtrdi 2280	. . . . . . 7 |- ( f : A -1-1-onto-> 2o -> ran f = { (/) , 1o } )
8	7	imaeq2d 5082	. . . . . 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 2278	. . . 4 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> dom f = ( `' f " { (/) , 1o } ) )
11		f1odm 5596	. . . . 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 5605	. . . . . . 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 5593	. . . . . 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 6652	. . . . . 6 |- (/) e. 2o
18	17	a1i 9	. . . . 5 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> (/) e. 2o )
19		1lt2o 6653	. . . . . 6 |- 1o e. 2o
20	19	a1i 9	. . . . 5 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> 1o e. 2o )
21		fnimapr 5715	. . . . 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 2272	. . 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 5932	. . . . 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 5932	. . . . . 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 3753	. . . . . . 7 |- ( y = ( `' f ` 1o ) -> { ( `' f ` (/) ) , y } = { ( `' f ` (/) ) , ( `' f ` 1o ) } )
30	29	eqeq2d 2243	. . . . . 6 |- ( y = ( `' f ` 1o ) -> ( A = { ( `' f ` (/) ) , y } <-> A = { ( `' f ` (/) ) , ( `' f ` 1o ) } ) )
31	30	spcegv 2895	. . . . 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 3752	. . . . . . 7 |- ( x = ( `' f ` (/) ) -> { x , y } = { ( `' f ` (/) ) , y } )
34	33	eqeq2d 2243	. . . . . 6 |- ( x = ( `' f ` (/) ) -> ( A = { x , y } <-> A = { ( `' f ` (/) ) , y } ) )
35	34	exbidv 1873	. . . . 5 |- ( x = ( `' f ` (/) ) -> ( E. y A = { x , y } <-> E. y A = { ( `' f ` (/) ) , y } ) )
36	35	spcegv 2895	. . . 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 1947	1 |- ( A ~~ 2o -> E. x E. y A = { x , y } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2202   (/)c0 3496   {cpr 3674   class class class wbr 4093   `'ccnv 4730   dom cdm 4731   ran crn 4732   "cima 4734   Fun wfun 5327   Fn wfn 5328   -1-1-onto->wf1o 5332   ` cfv 5333   1oc1o 6618   2oc2o 6619   ~~ cen 6950

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1o 6625  df-2o 6626  df-en 6953

This theorem is referenced by:  en2m  7042  en2prde  7441  upgrex  16027  upgr1een  16048