MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  zorn2 Unicode version

Theorem zorn2 8101
Description: Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set  A (with an ordering relation  R) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zorn2lem1 8091 through zorn2lem7 8097; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 8097. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
Hypothesis
Ref Expression
zornn0.1  |-  A  e. 
_V
Assertion
Ref Expression
zorn2  |-  ( ( R  Po  A  /\  A. w ( ( w 
C_  A  /\  R  Or  w )  ->  E. x  e.  A  A. z  e.  w  ( z R x  \/  z  =  x ) ) )  ->  E. x  e.  A  A. y  e.  A  -.  x R y )
Distinct variable groups:    x, w, y, z, A    w, R, x, y, z

Proof of Theorem zorn2
StepHypRef Expression
1 zornn0.1 . . 3  |-  A  e. 
_V
2 numth3 8065 . . 3  |-  ( A  e.  _V  ->  A  e.  dom  card )
31, 2ax-mp 10 . 2  |-  A  e. 
dom  card
4 zorn2g 8098 . 2  |-  ( ( A  e.  dom  card  /\  R  Po  A  /\  A. w ( ( w 
C_  A  /\  R  Or  w )  ->  E. x  e.  A  A. z  e.  w  ( z R x  \/  z  =  x ) ) )  ->  E. x  e.  A  A. y  e.  A  -.  x R y )
53, 4mp3an1 1269 1  |-  ( ( R  Po  A  /\  A. w ( ( w 
C_  A  /\  R  Or  w )  ->  E. x  e.  A  A. z  e.  w  ( z R x  \/  z  =  x ) ) )  ->  E. x  e.  A  A. y  e.  A  -.  x R y )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    \/ wo 359    /\ wa 360   A.wal 1532    = wceq 1619    e. wcel 1621   A.wral 2518   E.wrex 2519   _Vcvv 2763    C_ wss 3127   class class class wbr 3997    Po wpo 4284    Or wor 4285   dom cdm 4661   cardccrd 7536
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-ac2 8057
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-suc 4370  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-iota 6225  df-riota 6272  df-recs 6356  df-en 6832  df-card 7540  df-ac 7711
  Copyright terms: Public domain W3C validator