MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  zorn Unicode version

Theorem zorn 7992
Description: Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. This theorem is equivalent to the Axiom of Choice. Theorem 6M of [Enderton] p. 151. See zorn2 7991 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.)
Hypothesis
Ref Expression
zornn0.1  |-  A  e. 
_V
Assertion
Ref Expression
zorn  |-  ( A. z ( ( z 
C_  A  /\ [ C.]  Or  z )  ->  U. z  e.  A )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
Distinct variable group:    x, y, z, A

Proof of Theorem zorn
StepHypRef Expression
1 zornn0.1 . . 3  |-  A  e. 
_V
2 numth3 7955 . . 3  |-  ( A  e.  _V  ->  A  e.  dom  card )
31, 2ax-mp 9 . 2  |-  A  e. 
dom  card
4 zorng 7989 . 2  |-  ( ( A  e.  dom  card  /\ 
A. z ( ( z  C_  A  /\ [ C.] 
Or  z )  ->  U. z  e.  A
) )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
53, 4mpan 648 1  |-  ( A. z ( ( z 
C_  A  /\ [ C.]  Or  z )  ->  U. z  e.  A )  ->  E. x  e.  A  A. y  e.  A  -.  x  C.  y )
Colors of variables: wff set class
Syntax hints:   -. wn 4    -> wi 5    /\ wa 357   A.wal 1521    e. wcel 1610   A.wral 2495   E.wrex 2496   _Vcvv 2712    C_ wss 3058    C. wpss 3059   U.cuni 3707    Or wor 4185   dom cdm 4559   [ C.] crpss 6102   cardccrd 7426
This theorem is referenced by:  alexsubALTlem2  17420
This theorem was proved from axioms:  ax-1 6  ax-2 7  ax-3 8  ax-mp 9  ax-5 1522  ax-6 1523  ax-7 1524  ax-gen 1525  ax-8 1612  ax-11 1613  ax-13 1614  ax-14 1615  ax-17 1617  ax-12o 1653  ax-10 1667  ax-9 1673  ax-4 1681  ax-16 1915  ax-ext 2222  ax-rep 4007  ax-sep 4017  ax-nul 4025  ax-pow 4061  ax-pr 4087  ax-un 4382  ax-ac2 7947
This theorem depends on definitions:  df-bi 176  df-or 358  df-an 359  df-3or 934  df-3an 935  df-tru 1309  df-ex 1527  df-nf 1529  df-sb 1872  df-eu 2106  df-mo 2107  df-clab 2228  df-cleq 2234  df-clel 2237  df-nfc 2362  df-ne 2402  df-ral 2499  df-rex 2500  df-reu 2501  df-rab 2502  df-v 2714  df-sbc 2907  df-csb 2990  df-dif 3061  df-un 3063  df-in 3065  df-ss 3069  df-pss 3071  df-nul 3343  df-if 3451  df-pw 3512  df-sn 3530  df-pr 3531  df-tp 3532  df-op 3533  df-uni 3708  df-int 3741  df-iun 3785  df-br 3901  df-opab 3955  df-mpt 3956  df-tr 3990  df-eprel 4177  df-id 4181  df-po 4186  df-so 4187  df-fr 4224  df-se 4225  df-we 4226  df-ord 4267  df-on 4268  df-suc 4270  df-xp 4573  df-rel 4574  df-cnv 4575  df-co 4576  df-dm 4577  df-rn 4578  df-res 4579  df-ima 4580  df-fun 4581  df-fn 4582  df-f 4583  df-f1 4584  df-fo 4585  df-f1o 4586  df-fv 4587  df-isom 4588  df-rpss 6103  df-iota 6117  df-riota 6164  df-recs 6248  df-en 6724  df-card 7430  df-ac 7601
  Copyright terms: Public domain W3C validator