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Theorem zorn2 8386
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 8376 through zorn2lem7 8382; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 8382. (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 8350 . . 3  |-  ( A  e.  _V  ->  A  e.  dom  card )
31, 2ax-mp 8 . 2  |-  A  e. 
dom  card
4 zorn2g 8383 . 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 1266 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 3    -> wi 4    \/ wo 358    /\ wa 359   A.wal 1549    e. wcel 1725   A.wral 2705   E.wrex 2706   _Vcvv 2956    C_ wss 3320   class class class wbr 4212    Po wpo 4501    Or wor 4502   dom cdm 4878   cardccrd 7822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-ac2 8343
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-riota 6549  df-recs 6633  df-en 7110  df-card 7826  df-ac 7997
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