Theorem 0ct 7173

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 6498	. . . . 5 |- (/) e. 1o
2		djurcl 7118	. . . . 5 |- ( (/) e. 1o -> (inr ` (/) ) e. ( (/) ⊔ 1o ) )
3	1, 2	ax-mp 5	. . . 4 |- (inr ` (/) ) e. ( (/) ⊔ 1o )
4	3	fconst6 5457	. . 3 |- ( om X. { (inr ` (/) ) } ) : om --> ( (/) ⊔ 1o )
5		peano1 4630	. . . . 5 |- (/) e. om
6		rex0 3468	. . . . . . . . 9 |- -. E. w e. (/) y = (inl ` w )
7		djur 7135	. . . . . . . . . . 11 |- ( y e. ( (/) ⊔ 1o ) <-> ( E. w e. (/) y = (inl ` w ) \/ E. w e. 1o y = (inr ` w ) ) )
8	7	biimpi 120	. . . . . . . . . 10 |- ( y e. ( (/) ⊔ 1o ) -> ( E. w e. (/) y = (inl ` w ) \/ E. w e. 1o y = (inr ` w ) ) )
9	8	ord 725	. . . . . . . . 9 |- ( y e. ( (/) ⊔ 1o ) -> ( -. E. w e. (/) y = (inl ` w ) -> E. w e. 1o y = (inr ` w ) ) )
10	6, 9	mpi 15	. . . . . . . 8 |- ( y e. ( (/) ⊔ 1o ) -> E. w e. 1o y = (inr ` w ) )
11		df1o2 6487	. . . . . . . . 9 |- 1o = { (/) }
12	11	rexeqi 2698	. . . . . . . 8 |- ( E. w e. 1o y = (inr ` w ) <-> E. w e. { (/) } y = (inr ` w ) )
13	10, 12	sylib 122	. . . . . . 7 |- ( y e. ( (/) ⊔ 1o ) -> E. w e. { (/) } y = (inr ` w ) )
14		0ex 4160	. . . . . . . 8 |- (/) e. _V
15		fveq2 5558	. . . . . . . . 9 |- ( w = (/) -> (inr ` w ) = (inr ` (/) ) )
16	15	eqeq2d 2208	. . . . . . . 8 |- ( w = (/) -> ( y = (inr ` w ) <-> y = (inr ` (/) ) ) )
17	14, 16	rexsn 3666	. . . . . . 7 |- ( E. w e. { (/) } y = (inr ` w ) <-> y = (inr ` (/) ) )
18	13, 17	sylib 122	. . . . . 6 |- ( y e. ( (/) ⊔ 1o ) -> y = (inr ` (/) ) )
19	3	elexi 2775	. . . . . . . 8 |- (inr ` (/) ) e. _V
20	19	fvconst2 5778	. . . . . . 7 |- ( (/) e. om -> ( ( om X. { (inr ` (/) ) } ) ` (/) ) = (inr ` (/) ) )
21	5, 20	ax-mp 5	. . . . . 6 |- ( ( om X. { (inr ` (/) ) } ) ` (/) ) = (inr ` (/) )
22	18, 21	eqtr4di 2247	. . . . 5 |- ( y e. ( (/) ⊔ 1o ) -> y = ( ( om X. { (inr ` (/) ) } ) ` (/) ) )
23		fveq2 5558	. . . . . 6 |- ( z = (/) -> ( ( om X. { (inr ` (/) ) } ) ` z ) = ( ( om X. { (inr ` (/) ) } ) ` (/) ) )
24	23	rspceeqv 2886	. . . . 5 |- ( ( (/) e. om /\ y = ( ( om X. { (inr ` (/) ) } ) ` (/) ) ) -> E. z e. om y = ( ( om X. { (inr ` (/) ) } ) ` z ) )
25	5, 22, 24	sylancr 414	. . . 4 |- ( y e. ( (/) ⊔ 1o ) -> E. z e. om y = ( ( om X. { (inr ` (/) ) } ) ` z ) )
26	25	rgen 2550	. . 3 |- A. y e. ( (/) ⊔ 1o ) E. z e. om y = ( ( om X. { (inr ` (/) ) } ) ` z )
27		dffo3 5709	. . 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 ) ) )
28	4, 26, 27	mpbir2an 944	. 2 |- ( om X. { (inr ` (/) ) } ) : om -onto-> ( (/) ⊔ 1o )
29		omex 4629	. . . 4 |- om e. _V
30	19	snex 4218	. . . 4 |- { (inr ` (/) ) } e. _V
31	29, 30	xpex 4778	. . 3 |- ( om X. { (inr ` (/) ) } ) e. _V
32		foeq1 5476	. . 3 |- ( f = ( om X. { (inr ` (/) ) } ) -> ( f : om -onto-> ( (/) ⊔ 1o ) <-> ( om X. { (inr ` (/) ) } ) : om -onto-> ( (/) ⊔ 1o ) ) )
33	31, 32	spcev 2859	. 2 |- ( ( om X. { (inr ` (/) ) } ) : om -onto-> ( (/) ⊔ 1o ) -> E. f f : om -onto-> ( (/) ⊔ 1o ) )
34	28, 33	ax-mp 5	1 |- E. f f : om -onto-> ( (/) ⊔ 1o )

Syntax hints:   -. wn 3   \/ wo 709   = wceq 1364   E.wex 1506   e. wcel 2167   A.wral 2475   E.wrex 2476   (/)c0 3450   {csn 3622   omcom 4626   X. cxp 4661   -->wf 5254   -onto->wfo 5256   ` cfv 5258   1oc1o 6467   ⊔ cdju 7103  inlcinl 7111  inrcinr 7112

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199  df-1o 6474  df-dju 7104  df-inl 7113  df-inr 7114

This theorem is referenced by:  enumct  7181