Theorem 0ct 7108

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 6443	. . . . 5 |- (/) e. 1o
2		djurcl 7053	. . . . 5 |- ( (/) e. 1o -> (inr ` (/) ) e. ( (/) ⊔ 1o ) )
3	1, 2	ax-mp 5	. . . 4 |- (inr ` (/) ) e. ( (/) ⊔ 1o )
4	3	fconst6 5417	. . 3 |- ( om X. { (inr ` (/) ) } ) : om --> ( (/) ⊔ 1o )
5		peano1 4595	. . . . 5 |- (/) e. om
6		rex0 3442	. . . . . . . . 9 |- -. E. w e. (/) y = (inl ` w )
7		djur 7070	. . . . . . . . . . 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 724	. . . . . . . . 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 6432	. . . . . . . . 9 |- 1o = { (/) }
12	11	rexeqi 2678	. . . . . . . 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 4132	. . . . . . . 8 |- (/) e. _V
15		fveq2 5517	. . . . . . . . 9 |- ( w = (/) -> (inr ` w ) = (inr ` (/) ) )
16	15	eqeq2d 2189	. . . . . . . 8 |- ( w = (/) -> ( y = (inr ` w ) <-> y = (inr ` (/) ) ) )
17	14, 16	rexsn 3638	. . . . . . 7 |- ( E. w e. { (/) } y = (inr ` w ) <-> y = (inr ` (/) ) )
18	13, 17	sylib 122	. . . . . 6 |- ( y e. ( (/) ⊔ 1o ) -> y = (inr ` (/) ) )
19	3	elexi 2751	. . . . . . . 8 |- (inr ` (/) ) e. _V
20	19	fvconst2 5734	. . . . . . 7 |- ( (/) e. om -> ( ( om X. { (inr ` (/) ) } ) ` (/) ) = (inr ` (/) ) )
21	5, 20	ax-mp 5	. . . . . 6 |- ( ( om X. { (inr ` (/) ) } ) ` (/) ) = (inr ` (/) )
22	18, 21	eqtr4di 2228	. . . . 5 |- ( y e. ( (/) ⊔ 1o ) -> y = ( ( om X. { (inr ` (/) ) } ) ` (/) ) )
23		fveq2 5517	. . . . . 6 |- ( z = (/) -> ( ( om X. { (inr ` (/) ) } ) ` z ) = ( ( om X. { (inr ` (/) ) } ) ` (/) ) )
24	23	rspceeqv 2861	. . . . 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 2530	. . 3 |- A. y e. ( (/) ⊔ 1o ) E. z e. om y = ( ( om X. { (inr ` (/) ) } ) ` z )
27		dffo3 5665	. . 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 942	. 2 |- ( om X. { (inr ` (/) ) } ) : om -onto-> ( (/) ⊔ 1o )
29		omex 4594	. . . 4 |- om e. _V
30	19	snex 4187	. . . 4 |- { (inr ` (/) ) } e. _V
31	29, 30	xpex 4743	. . 3 |- ( om X. { (inr ` (/) ) } ) e. _V
32		foeq1 5436	. . 3 |- ( f = ( om X. { (inr ` (/) ) } ) -> ( f : om -onto-> ( (/) ⊔ 1o ) <-> ( om X. { (inr ` (/) ) } ) : om -onto-> ( (/) ⊔ 1o ) ) )
33	31, 32	spcev 2834	. 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 708   = wceq 1353   E.wex 1492   e. wcel 2148   A.wral 2455   E.wrex 2456   (/)c0 3424   {csn 3594   omcom 4591   X. cxp 4626   -->wf 5214   -onto->wfo 5216   ` cfv 5218   1oc1o 6412   ⊔ cdju 7038  inlcinl 7046  inrcinr 7047

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-1o 6419  df-dju 7039  df-inl 7048  df-inr 7049

This theorem is referenced by:  enumct  7116