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Theorem ttukey 8675
Description: The Teichmüller-Tukey Lemma, an Axiom of Choice equivalent. If  A is a nonempty collection of finite character, then  A has a maximal element with respect to inclusion. Here "finite character" means that  x  e.  A iff every finite subset of  x is in  A. (Contributed by Mario Carneiro, 15-May-2015.)
Hypothesis
Ref Expression
ttukey.1  |-  A  e. 
_V
Assertion
Ref Expression
ttukey  |-  ( ( A  =/=  (/)  /\  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A
) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
Distinct variable group:    x, y, A

Proof of Theorem ttukey
StepHypRef Expression
1 ttukey.1 . . . 4  |-  A  e. 
_V
21uniex 6365 . . 3  |-  U. A  e.  _V
3 numth3 8627 . . 3  |-  ( U. A  e.  _V  ->  U. A  e.  dom  card )
42, 3ax-mp 5 . 2  |-  U. A  e.  dom  card
5 ttukeyg 8674 . 2  |-  ( ( U. A  e.  dom  card  /\  A  =/=  (/)  /\  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A
) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
64, 5mp3an1 1294 1  |-  ( ( A  =/=  (/)  /\  A. x ( x  e.  A  <->  ( ~P x  i^i  Fin )  C_  A
) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1360    e. wcel 1755    =/= wne 2596   A.wral 2705   E.wrex 2706   _Vcvv 2962    i^i cin 3315    C_ wss 3316    C. wpss 3317   (/)c0 3625   ~Pcpw 3848   U.cuni 4079   dom cdm 4827   Fincfn 7298   cardccrd 8093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-ac2 8620
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-om 6466  df-recs 6818  df-1o 6908  df-er 7089  df-en 7299  df-dom 7300  df-fin 7302  df-card 8097  df-ac 8274
This theorem is referenced by: (None)
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