Theorem 0ct 6869

Description: The empty set is countable. Remark of [BauerSwan], p. 14:3 which also has the definition of countable used here. (Contributed by Jim Kingdon, 13-Mar-2023.)

Assertion

Ref	Expression
0ct	|- E. f f : om -onto-> ( (/) ⊔ 1o )

Proof of Theorem 0ct

Dummy variables y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0lt1o 6242	. . . . 5 |- (/) e. 1o
2		djurcl 6824	. . . . 5 |- ( (/) e. 1o -> (inr ` (/) ) e. ( (/) ⊔ 1o ) )
3	1, 2	ax-mp 7	. . . 4 |- (inr ` (/) ) e. ( (/) ⊔ 1o )
4	3	fconst6 5245	. . 3 |- ( om X. { (inr ` (/) ) } ) : om --> ( (/) ⊔ 1o )
5		peano1 4437	. . . . 5 |- (/) e. om
6		rex0 3319	. . . . . . . . 9 |- -. E. w e. (/) y = (inl ` w )
7		djur 6837	. . . . . . . . . 10 |- ( y e. ( (/) ⊔ 1o ) -> ( E. w e. (/) y = (inl ` w ) \/ E. w e. 1o y = (inr ` w ) ) )
8	7	ord 681	. . . . . . . . 9 |- ( y e. ( (/) ⊔ 1o ) -> ( -. E. w e. (/) y = (inl ` w ) -> E. w e. 1o y = (inr ` w ) ) )
9	6, 8	mpi 15	. . . . . . . 8 |- ( y e. ( (/) ⊔ 1o ) -> E. w e. 1o y = (inr ` w ) )
10		df1o2 6232	. . . . . . . . 9 |- 1o = { (/) }
11	10	rexeqi 2581	. . . . . . . 8 |- ( E. w e. 1o y = (inr ` w ) <-> E. w e. { (/) } y = (inr ` w ) )
12	9, 11	sylib 121	. . . . . . 7 |- ( y e. ( (/) ⊔ 1o ) -> E. w e. { (/) } y = (inr ` w ) )
13		0ex 3987	. . . . . . . 8 |- (/) e. _V
14		fveq2 5340	. . . . . . . . 9 |- ( w = (/) -> (inr ` w ) = (inr ` (/) ) )
15	14	eqeq2d 2106	. . . . . . . 8 |- ( w = (/) -> ( y = (inr ` w ) <-> y = (inr ` (/) ) ) )
16	13, 15	rexsn 3507	. . . . . . 7 |- ( E. w e. { (/) } y = (inr ` w ) <-> y = (inr ` (/) ) )
17	12, 16	sylib 121	. . . . . 6 |- ( y e. ( (/) ⊔ 1o ) -> y = (inr ` (/) ) )
18	3	elexi 2645	. . . . . . . 8 |- (inr ` (/) ) e. _V
19	18	fvconst2 5552	. . . . . . 7 |- ( (/) e. om -> ( ( om X. { (inr ` (/) ) } ) ` (/) ) = (inr ` (/) ) )
20	5, 19	ax-mp 7	. . . . . 6 |- ( ( om X. { (inr ` (/) ) } ) ` (/) ) = (inr ` (/) )
21	17, 20	syl6eqr 2145	. . . . 5 |- ( y e. ( (/) ⊔ 1o ) -> y = ( ( om X. { (inr ` (/) ) } ) ` (/) ) )
22		fveq2 5340	. . . . . 6 |- ( z = (/) -> ( ( om X. { (inr ` (/) ) } ) ` z ) = ( ( om X. { (inr ` (/) ) } ) ` (/) ) )
23	22	rspceeqv 2753	. . . . 5 |- ( ( (/) e. om /\ y = ( ( om X. { (inr ` (/) ) } ) ` (/) ) ) -> E. z e. om y = ( ( om X. { (inr ` (/) ) } ) ` z ) )
24	5, 21, 23	sylancr 406	. . . 4 |- ( y e. ( (/) ⊔ 1o ) -> E. z e. om y = ( ( om X. { (inr ` (/) ) } ) ` z ) )
25	24	rgen 2439	. . 3 |- A. y e. ( (/) ⊔ 1o ) E. z e. om y = ( ( om X. { (inr ` (/) ) } ) ` z )
26		dffo3 5485	. . 3 |- ( ( om X. { (inr ` (/) ) } ) : om -onto-> ( (/) ⊔ 1o ) <-> ( ( om X. { (inr ` (/) ) } ) : om --> ( (/) ⊔ 1o ) /\ A. y e. ( (/) ⊔ 1o ) E. z e. om y = ( ( om X. { (inr ` (/) ) } ) ` z ) ) )
27	4, 25, 26	mpbir2an 891	. 2 |- ( om X. { (inr ` (/) ) } ) : om -onto-> ( (/) ⊔ 1o )
28		omex 4436	. . . 4 |- om e. _V
29	18	snex 4041	. . . 4 |- { (inr ` (/) ) } e. _V
30	28, 29	xpex 4582	. . 3 |- ( om X. { (inr ` (/) ) } ) e. _V
31		foeq1 5264	. . 3 |- ( f = ( om X. { (inr ` (/) ) } ) -> ( f : om -onto-> ( (/) ⊔ 1o ) <-> ( om X. { (inr ` (/) ) } ) : om -onto-> ( (/) ⊔ 1o ) ) )
32	30, 31	spcev 2727	. 2 |- ( ( om X. { (inr ` (/) ) } ) : om -onto-> ( (/) ⊔ 1o ) -> E. f f : om -onto-> ( (/) ⊔ 1o ) )
33	27, 32	ax-mp 7	1 |- E. f f : om -onto-> ( (/) ⊔ 1o )

Syntax hints:   -. wn 3   = wceq 1296   E.wex 1433   e. wcel 1445   A.wral 2370   E.wrex 2371   (/)c0 3302   {csn 3466   omcom 4433   X. cxp 4465   -->wf 5045   -onto->wfo 5047   ` cfv 5049   1oc1o 6212   ⊔ cdju 6810  inlcinl 6817  inrcinr 6818

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fo 5055  df-fv 5057  df-1st 5949  df-2nd 5950  df-1o 6219  df-dju 6811  df-inl 6819  df-inr 6820

This theorem is referenced by:  enumct  6873