Theorem 0ct 7072

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 6408	. . . . 5 |- (/) e. 1o
2		djurcl 7017	. . . . 5 |- ( (/) e. 1o -> (inr ` (/) ) e. ( (/) ⊔ 1o ) )
3	1, 2	ax-mp 5	. . . 4 |- (inr ` (/) ) e. ( (/) ⊔ 1o )
4	3	fconst6 5387	. . 3 |- ( om X. { (inr ` (/) ) } ) : om --> ( (/) ⊔ 1o )
5		peano1 4571	. . . . 5 |- (/) e. om
6		rex0 3426	. . . . . . . . 9 |- -. E. w e. (/) y = (inl ` w )
7		djur 7034	. . . . . . . . . . 11 |- ( y e. ( (/) ⊔ 1o ) <-> ( E. w e. (/) y = (inl ` w ) \/ E. w e. 1o y = (inr ` w ) ) )
8	7	biimpi 119	. . . . . . . . . 10 |- ( y e. ( (/) ⊔ 1o ) -> ( E. w e. (/) y = (inl ` w ) \/ E. w e. 1o y = (inr ` w ) ) )
9	8	ord 714	. . . . . . . . 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 6397	. . . . . . . . 9 |- 1o = { (/) }
12	11	rexeqi 2666	. . . . . . . 8 |- ( E. w e. 1o y = (inr ` w ) <-> E. w e. { (/) } y = (inr ` w ) )
13	10, 12	sylib 121	. . . . . . 7 |- ( y e. ( (/) ⊔ 1o ) -> E. w e. { (/) } y = (inr ` w ) )
14		0ex 4109	. . . . . . . 8 |- (/) e. _V
15		fveq2 5486	. . . . . . . . 9 |- ( w = (/) -> (inr ` w ) = (inr ` (/) ) )
16	15	eqeq2d 2177	. . . . . . . 8 |- ( w = (/) -> ( y = (inr ` w ) <-> y = (inr ` (/) ) ) )
17	14, 16	rexsn 3620	. . . . . . 7 |- ( E. w e. { (/) } y = (inr ` w ) <-> y = (inr ` (/) ) )
18	13, 17	sylib 121	. . . . . 6 |- ( y e. ( (/) ⊔ 1o ) -> y = (inr ` (/) ) )
19	3	elexi 2738	. . . . . . . 8 |- (inr ` (/) ) e. _V
20	19	fvconst2 5701	. . . . . . 7 |- ( (/) e. om -> ( ( om X. { (inr ` (/) ) } ) ` (/) ) = (inr ` (/) ) )
21	5, 20	ax-mp 5	. . . . . 6 |- ( ( om X. { (inr ` (/) ) } ) ` (/) ) = (inr ` (/) )
22	18, 21	eqtr4di 2217	. . . . 5 |- ( y e. ( (/) ⊔ 1o ) -> y = ( ( om X. { (inr ` (/) ) } ) ` (/) ) )
23		fveq2 5486	. . . . . 6 |- ( z = (/) -> ( ( om X. { (inr ` (/) ) } ) ` z ) = ( ( om X. { (inr ` (/) ) } ) ` (/) ) )
24	23	rspceeqv 2848	. . . . 5 |- ( ( (/) e. om /\ y = ( ( om X. { (inr ` (/) ) } ) ` (/) ) ) -> E. z e. om y = ( ( om X. { (inr ` (/) ) } ) ` z ) )
25	5, 22, 24	sylancr 411	. . . 4 |- ( y e. ( (/) ⊔ 1o ) -> E. z e. om y = ( ( om X. { (inr ` (/) ) } ) ` z ) )
26	25	rgen 2519	. . 3 |- A. y e. ( (/) ⊔ 1o ) E. z e. om y = ( ( om X. { (inr ` (/) ) } ) ` z )
27		dffo3 5632	. . 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 932	. 2 |- ( om X. { (inr ` (/) ) } ) : om -onto-> ( (/) ⊔ 1o )
29		omex 4570	. . . 4 |- om e. _V
30	19	snex 4164	. . . 4 |- { (inr ` (/) ) } e. _V
31	29, 30	xpex 4719	. . 3 |- ( om X. { (inr ` (/) ) } ) e. _V
32		foeq1 5406	. . 3 |- ( f = ( om X. { (inr ` (/) ) } ) -> ( f : om -onto-> ( (/) ⊔ 1o ) <-> ( om X. { (inr ` (/) ) } ) : om -onto-> ( (/) ⊔ 1o ) ) )
33	31, 32	spcev 2821	. 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 698   = wceq 1343   E.wex 1480   e. wcel 2136   A.wral 2444   E.wrex 2445   (/)c0 3409   {csn 3576   omcom 4567   X. cxp 4602   -->wf 5184   -onto->wfo 5186   ` cfv 5188   1oc1o 6377   ⊔ cdju 7002  inlcinl 7010  inrcinr 7011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-1o 6384  df-dju 7003  df-inl 7012  df-inr 7013

This theorem is referenced by:  enumct  7080