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Theorem zorn2 8936
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 8926 through zorn2lem7 8932; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 8932. (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 8900 . . 3  |-  ( A  e.  _V  ->  A  e.  dom  card )
31, 2ax-mp 5 . 2  |-  A  e. 
dom  card
4 zorn2g 8933 . 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 1351 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 370    /\ wa 371   A.wal 1442    e. wcel 1887   A.wral 2737   E.wrex 2738   _Vcvv 3045    C_ wss 3404   class class class wbr 4402    Po wpo 4753    Or wor 4754   dom cdm 4834   cardccrd 8369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-ac2 8893
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-wrecs 7028  df-recs 7090  df-en 7570  df-card 8373  df-ac 8547
This theorem is referenced by: (None)
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