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Theorem 0ct 6985
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.
StepHypRef Expression
1 0lt1o 6330 . . . . 5  |-  (/)  e.  1o
2 djurcl 6930 . . . . 5  |-  ( (/)  e.  1o  ->  (inr `  (/) )  e.  ( (/) 1o ) )
31, 2ax-mp 5 . . . 4  |-  (inr `  (/) )  e.  ( (/) 1o )
43fconst6 5317 . . 3  |-  ( om 
X.  { (inr `  (/) ) } ) : om --> ( (/) 1o )
5 peano1 4503 . . . . 5  |-  (/)  e.  om
6 rex0 3375 . . . . . . . . 9  |-  -.  E. w  e.  (/)  y  =  (inl `  w )
7 djur 6947 . . . . . . . . . . 11  |-  ( y  e.  ( (/) 1o )  <-> 
( E. w  e.  (/)  y  =  (inl `  w )  \/  E. w  e.  1o  y  =  (inr `  w )
) )
87biimpi 119 . . . . . . . . . 10  |-  ( y  e.  ( (/) 1o )  ->  ( E. w  e.  (/)  y  =  (inl
`  w )  \/ 
E. w  e.  1o  y  =  (inr `  w
) ) )
98ord 713 . . . . . . . . 9  |-  ( y  e.  ( (/) 1o )  ->  ( -.  E. w  e.  (/)  y  =  (inl `  w )  ->  E. w  e.  1o  y  =  (inr `  w
) ) )
106, 9mpi 15 . . . . . . . 8  |-  ( y  e.  ( (/) 1o )  ->  E. w  e.  1o  y  =  (inr `  w
) )
11 df1o2 6319 . . . . . . . . 9  |-  1o  =  { (/) }
1211rexeqi 2629 . . . . . . . 8  |-  ( E. w  e.  1o  y  =  (inr `  w )  <->  E. w  e.  { (/) } y  =  (inr `  w ) )
1310, 12sylib 121 . . . . . . 7  |-  ( y  e.  ( (/) 1o )  ->  E. w  e.  { (/)
} y  =  (inr
`  w ) )
14 0ex 4050 . . . . . . . 8  |-  (/)  e.  _V
15 fveq2 5414 . . . . . . . . 9  |-  ( w  =  (/)  ->  (inr `  w )  =  (inr
`  (/) ) )
1615eqeq2d 2149 . . . . . . . 8  |-  ( w  =  (/)  ->  ( y  =  (inr `  w
)  <->  y  =  (inr
`  (/) ) ) )
1714, 16rexsn 3563 . . . . . . 7  |-  ( E. w  e.  { (/) } y  =  (inr `  w )  <->  y  =  (inr `  (/) ) )
1813, 17sylib 121 . . . . . 6  |-  ( y  e.  ( (/) 1o )  ->  y  =  (inr
`  (/) ) )
193elexi 2693 . . . . . . . 8  |-  (inr `  (/) )  e.  _V
2019fvconst2 5629 . . . . . . 7  |-  ( (/)  e.  om  ->  ( ( om  X.  { (inr `  (/) ) } ) `  (/) )  =  (inr `  (/) ) )
215, 20ax-mp 5 . . . . . 6  |-  ( ( om  X.  { (inr
`  (/) ) } ) `
 (/) )  =  (inr
`  (/) )
2218, 21syl6eqr 2188 . . . . 5  |-  ( y  e.  ( (/) 1o )  ->  y  =  ( ( om  X.  {
(inr `  (/) ) } ) `  (/) ) )
23 fveq2 5414 . . . . . 6  |-  ( z  =  (/)  ->  ( ( om  X.  { (inr
`  (/) ) } ) `
 z )  =  ( ( om  X.  { (inr `  (/) ) } ) `  (/) ) )
2423rspceeqv 2802 . . . . 5  |-  ( (
(/)  e.  om  /\  y  =  ( ( om 
X.  { (inr `  (/) ) } ) `  (/) ) )  ->  E. z  e.  om  y  =  ( ( om  X.  {
(inr `  (/) ) } ) `  z ) )
255, 22, 24sylancr 410 . . . 4  |-  ( y  e.  ( (/) 1o )  ->  E. z  e.  om  y  =  ( ( om  X.  { (inr `  (/) ) } ) `  z ) )
2625rgen 2483 . . 3  |-  A. y  e.  ( (/) 1o ) E. z  e.  om  y  =  ( ( om 
X.  { (inr `  (/) ) } ) `  z )
27 dffo3 5560 . . 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 ) ) )
284, 26, 27mpbir2an 926 . 2  |-  ( om 
X.  { (inr `  (/) ) } ) : om -onto-> ( (/) 1o )
29 omex 4502 . . . 4  |-  om  e.  _V
3019snex 4104 . . . 4  |-  { (inr
`  (/) ) }  e.  _V
3129, 30xpex 4649 . . 3  |-  ( om 
X.  { (inr `  (/) ) } )  e. 
_V
32 foeq1 5336 . . 3  |-  ( f  =  ( om  X.  { (inr `  (/) ) } )  ->  ( f : om -onto-> ( (/) 1o )  <-> 
( om  X.  {
(inr `  (/) ) } ) : om -onto-> ( (/) 1o ) ) )
3331, 32spcev 2775 . 2  |-  ( ( om  X.  { (inr
`  (/) ) } ) : om -onto-> ( (/) 1o )  ->  E. f 
f : om -onto-> ( (/) 1o ) )
3428, 33ax-mp 5 1  |-  E. f 
f : om -onto-> ( (/) 1o )
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 697    = wceq 1331   E.wex 1468    e. wcel 1480   A.wral 2414   E.wrex 2415   (/)c0 3358   {csn 3522   omcom 4499    X. cxp 4532   -->wf 5114   -onto->wfo 5116   ` cfv 5118   1oc1o 6299   ⊔ cdju 6915  inlcinl 6923  inrcinr 6924
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-2nd 6032  df-1o 6306  df-dju 6916  df-inl 6925  df-inr 6926
This theorem is referenced by:  enumct  6993
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