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Theorem pw1fin 6888
Description: Excluded middle is equivalent to the power set of  1o being finite. (Contributed by SN and Jim Kingdon, 7-Aug-2024.)
Assertion
Ref Expression
pw1fin  |-  (EXMID  <->  ~P 1o  e.  Fin )

Proof of Theorem pw1fin
StepHypRef Expression
1 exmidpweq 6887 . . . 4  |-  (EXMID  <->  ~P 1o  =  2o )
21biimpi 119 . . 3  |-  (EXMID  ->  ~P 1o  =  2o )
3 2onn 6500 . . . 4  |-  2o  e.  om
4 nnfi 6850 . . . 4  |-  ( 2o  e.  om  ->  2o  e.  Fin )
53, 4ax-mp 5 . . 3  |-  2o  e.  Fin
62, 5eqeltrdi 2261 . 2  |-  (EXMID  ->  ~P 1o  e.  Fin )
7 df1o2 6408 . . . . . 6  |-  1o  =  { (/) }
87sseq2i 3174 . . . . 5  |-  ( x 
C_  1o  <->  x  C_  { (/) } )
9 velpw 3573 . . . . . 6  |-  ( x  e.  ~P 1o  <->  x  C_  1o )
10 1oex 6403 . . . . . . . 8  |-  1o  e.  _V
1110pwid 3581 . . . . . . 7  |-  1o  e.  ~P 1o
12 fidceq 6847 . . . . . . 7  |-  ( ( ~P 1o  e.  Fin  /\  x  e.  ~P 1o  /\  1o  e.  ~P 1o )  -> DECID 
x  =  1o )
1311, 12mp3an3 1321 . . . . . 6  |-  ( ( ~P 1o  e.  Fin  /\  x  e.  ~P 1o )  -> DECID 
x  =  1o )
149, 13sylan2br 286 . . . . 5  |-  ( ( ~P 1o  e.  Fin  /\  x  C_  1o )  -> DECID  x  =  1o )
158, 14sylan2br 286 . . . 4  |-  ( ( ~P 1o  e.  Fin  /\  x  C_  { (/) } )  -> DECID 
x  =  1o )
167eqeq2i 2181 . . . . 5  |-  ( x  =  1o  <->  x  =  { (/) } )
1716dcbii 835 . . . 4  |-  (DECID  x  =  1o  <-> DECID  x  =  { (/) } )
1815, 17sylib 121 . . 3  |-  ( ( ~P 1o  e.  Fin  /\  x  C_  { (/) } )  -> DECID 
x  =  { (/) } )
1918exmid1dc 4186 . 2  |-  ( ~P 1o  e.  Fin  -> EXMID )
206, 19impbii 125 1  |-  (EXMID  <->  ~P 1o  e.  Fin )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104  DECID wdc 829    = wceq 1348    e. wcel 2141    C_ wss 3121   (/)c0 3414   ~Pcpw 3566   {csn 3583  EXMIDwem 4180   omcom 4574   1oc1o 6388   2oc2o 6389   Fincfn 6718
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-tr 4088  df-exmid 4181  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1o 6395  df-2o 6396  df-en 6719  df-fin 6721
This theorem is referenced by: (None)
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