Theorem en2 6926

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 6848	. . 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 5055	. . . . 5 |- ( `' f " ran f ) = dom f
4		dff1o2 5539	. . . . . . . . 9 |- ( f : A -1-1-onto-> 2o <-> ( f Fn A /\ Fun `' f /\ ran f = 2o ) )
5	4	simp3bi 1017	. . . . . . . 8 |- ( f : A -1-1-onto-> 2o -> ran f = 2o )
6		df2o3 6529	. . . . . . . 8 |- 2o = { (/) , 1o }
7	5, 6	eqtrdi 2255	. . . . . . 7 |- ( f : A -1-1-onto-> 2o -> ran f = { (/) , 1o } )
8	7	imaeq2d 5031	. . . . . 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 2253	. . . 4 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> dom f = ( `' f " { (/) , 1o } ) )
11		f1odm 5538	. . . . 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 5547	. . . . . . 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 5535	. . . . . 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 6540	. . . . . 6 |- (/) e. 2o
18	17	a1i 9	. . . . 5 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> (/) e. 2o )
19		1lt2o 6541	. . . . . 6 |- 1o e. 2o
20	19	a1i 9	. . . . 5 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> 1o e. 2o )
21		fnimapr 5652	. . . . 5 |- ( ( `' f Fn 2o /\ (/) e. 2o /\ 1o e. 2o ) -> ( `' f " { (/) , 1o } ) = { ( `' f ` (/) ) , ( `' f ` 1o ) } )
22	16, 18, 20, 21	syl3anc 1250	. . . 4 |- ( ( A ~~ 2o /\ f : A -1-1-onto-> 2o ) -> ( `' f " { (/) , 1o } ) = { ( `' f ` (/) ) , ( `' f ` 1o ) } )
23	10, 12, 22	3eqtr3d 2247	. . 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 5863	. . . . 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 5863	. . . . . 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 3716	. . . . . . 7 |- ( y = ( `' f ` 1o ) -> { ( `' f ` (/) ) , y } = { ( `' f ` (/) ) , ( `' f ` 1o ) } )
30	29	eqeq2d 2218	. . . . . 6 |- ( y = ( `' f ` 1o ) -> ( A = { ( `' f ` (/) ) , y } <-> A = { ( `' f ` (/) ) , ( `' f ` 1o ) } ) )
31	30	spcegv 2865	. . . . 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 3715	. . . . . . 7 |- ( x = ( `' f ` (/) ) -> { x , y } = { ( `' f ` (/) ) , y } )
34	33	eqeq2d 2218	. . . . . 6 |- ( x = ( `' f ` (/) ) -> ( A = { x , y } <-> A = { ( `' f ` (/) ) , y } ) )
35	34	exbidv 1849	. . . . 5 |- ( x = ( `' f ` (/) ) -> ( E. y A = { x , y } <-> E. y A = { ( `' f ` (/) ) , y } ) )
36	35	spcegv 2865	. . . 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 1923	1 |- ( A ~~ 2o -> E. x E. y A = { x , y } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   E.wex 1516   e. wcel 2177   (/)c0 3464   {cpr 3639   class class class wbr 4051   `'ccnv 4682   dom cdm 4683   ran crn 4684   "cima 4686   Fun wfun 5274   Fn wfn 5275   -1-1-onto->wf1o 5279   ` cfv 5280   1oc1o 6508   2oc2o 6509   ~~ cen 6838

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-suc 4426  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-1o 6515  df-2o 6516  df-en 6841

This theorem is referenced by:  en2m  6927  upgrex  15774