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Theorem 0ct 7120
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 6455 . . . . 5  |-  (/)  e.  1o
2 djurcl 7065 . . . . 5  |-  ( (/)  e.  1o  ->  (inr `  (/) )  e.  ( (/) 1o ) )
31, 2ax-mp 5 . . . 4  |-  (inr `  (/) )  e.  ( (/) 1o )
43fconst6 5427 . . 3  |-  ( om 
X.  { (inr `  (/) ) } ) : om --> ( (/) 1o )
5 peano1 4605 . . . . 5  |-  (/)  e.  om
6 rex0 3452 . . . . . . . . 9  |-  -.  E. w  e.  (/)  y  =  (inl `  w )
7 djur 7082 . . . . . . . . . . 11  |-  ( y  e.  ( (/) 1o )  <-> 
( E. w  e.  (/)  y  =  (inl `  w )  \/  E. w  e.  1o  y  =  (inr `  w )
) )
87biimpi 120 . . . . . . . . . 10  |-  ( y  e.  ( (/) 1o )  ->  ( E. w  e.  (/)  y  =  (inl
`  w )  \/ 
E. w  e.  1o  y  =  (inr `  w
) ) )
98ord 725 . . . . . . . . 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 6444 . . . . . . . . 9  |-  1o  =  { (/) }
1211rexeqi 2688 . . . . . . . 8  |-  ( E. w  e.  1o  y  =  (inr `  w )  <->  E. w  e.  { (/) } y  =  (inr `  w ) )
1310, 12sylib 122 . . . . . . 7  |-  ( y  e.  ( (/) 1o )  ->  E. w  e.  { (/)
} y  =  (inr
`  w ) )
14 0ex 4142 . . . . . . . 8  |-  (/)  e.  _V
15 fveq2 5527 . . . . . . . . 9  |-  ( w  =  (/)  ->  (inr `  w )  =  (inr
`  (/) ) )
1615eqeq2d 2199 . . . . . . . 8  |-  ( w  =  (/)  ->  ( y  =  (inr `  w
)  <->  y  =  (inr
`  (/) ) ) )
1714, 16rexsn 3648 . . . . . . 7  |-  ( E. w  e.  { (/) } y  =  (inr `  w )  <->  y  =  (inr `  (/) ) )
1813, 17sylib 122 . . . . . 6  |-  ( y  e.  ( (/) 1o )  ->  y  =  (inr
`  (/) ) )
193elexi 2761 . . . . . . . 8  |-  (inr `  (/) )  e.  _V
2019fvconst2 5745 . . . . . . 7  |-  ( (/)  e.  om  ->  ( ( om  X.  { (inr `  (/) ) } ) `  (/) )  =  (inr `  (/) ) )
215, 20ax-mp 5 . . . . . 6  |-  ( ( om  X.  { (inr
`  (/) ) } ) `
 (/) )  =  (inr
`  (/) )
2218, 21eqtr4di 2238 . . . . 5  |-  ( y  e.  ( (/) 1o )  ->  y  =  ( ( om  X.  {
(inr `  (/) ) } ) `  (/) ) )
23 fveq2 5527 . . . . . 6  |-  ( z  =  (/)  ->  ( ( om  X.  { (inr
`  (/) ) } ) `
 z )  =  ( ( om  X.  { (inr `  (/) ) } ) `  (/) ) )
2423rspceeqv 2871 . . . . 5  |-  ( (
(/)  e.  om  /\  y  =  ( ( om 
X.  { (inr `  (/) ) } ) `  (/) ) )  ->  E. z  e.  om  y  =  ( ( om  X.  {
(inr `  (/) ) } ) `  z ) )
255, 22, 24sylancr 414 . . . 4  |-  ( y  e.  ( (/) 1o )  ->  E. z  e.  om  y  =  ( ( om  X.  { (inr `  (/) ) } ) `  z ) )
2625rgen 2540 . . 3  |-  A. y  e.  ( (/) 1o ) E. z  e.  om  y  =  ( ( om 
X.  { (inr `  (/) ) } ) `  z )
27 dffo3 5676 . . 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 943 . 2  |-  ( om 
X.  { (inr `  (/) ) } ) : om -onto-> ( (/) 1o )
29 omex 4604 . . . 4  |-  om  e.  _V
3019snex 4197 . . . 4  |-  { (inr
`  (/) ) }  e.  _V
3129, 30xpex 4753 . . 3  |-  ( om 
X.  { (inr `  (/) ) } )  e. 
_V
32 foeq1 5446 . . 3  |-  ( f  =  ( om  X.  { (inr `  (/) ) } )  ->  ( f : om -onto-> ( (/) 1o )  <-> 
( om  X.  {
(inr `  (/) ) } ) : om -onto-> ( (/) 1o ) ) )
3331, 32spcev 2844 . 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 709    = wceq 1363   E.wex 1502    e. wcel 2158   A.wral 2465   E.wrex 2466   (/)c0 3434   {csn 3604   omcom 4601    X. cxp 4636   -->wf 5224   -onto->wfo 5226   ` cfv 5228   1oc1o 6424   ⊔ cdju 7050  inlcinl 7058  inrcinr 7059
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-1st 6155  df-2nd 6156  df-1o 6431  df-dju 7051  df-inl 7060  df-inr 7061
This theorem is referenced by:  enumct  7128
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