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Theorem zorn 7654
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 7653 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 7599 . . 3  |-  ( A  e.  _V  ->  A  e.  dom  card )
31, 2ax-mp 8 . 2  |-  A  e. 
dom  card
4 zorng 7651 . 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 646 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 3    -> wi 4    /\ wa 356   A.wal 1451    e. wcel 1533   A.wral 2311   E.wrex 2312   _Vcvv 2512    C_ wss 2830    C. wpss 2831   U.cuni 3468    Or wor 3908   dom cdm 4300   [ C.] crpss 5799   cardccrd 7094
This theorem is referenced by:  alexsubALTlem2  16484
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1452  ax-6 1453  ax-7 1454  ax-gen 1455  ax-8 1535  ax-11 1536  ax-13 1537  ax-14 1538  ax-17 1540  ax-12o 1574  ax-10 1588  ax-9 1594  ax-4 1601  ax-16 1787  ax-ext 2082  ax-rep 3735  ax-sep 3745  ax-nul 3753  ax-pow 3789  ax-pr 3813  ax-un 4105  ax-ac 7592
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 901  df-3an 902  df-ex 1457  df-sb 1748  df-eu 1970  df-mo 1971  df-clab 2088  df-cleq 2093  df-clel 2096  df-ne 2220  df-ral 2315  df-rex 2316  df-reu 2317  df-rab 2318  df-v 2514  df-sbc 2688  df-csb 2770  df-dif 2833  df-un 2835  df-in 2837  df-ss 2841  df-pss 2843  df-nul 3111  df-if 3221  df-pw 3282  df-sn 3300  df-pr 3301  df-tp 3302  df-op 3303  df-uni 3469  df-int 3503  df-iun 3546  df-br 3631  df-opab 3685  df-mpt 3686  df-tr 3718  df-eprel 3900  df-id 3904  df-po 3909  df-so 3910  df-fr 3947  df-se 3948  df-we 3949  df-ord 3990  df-on 3991  df-suc 3993  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-fun 4322  df-fn 4323  df-f 4324  df-f1 4325  df-fo 4326  df-f1o 4327  df-fv 4328  df-iso 4329  df-rpss 5800  df-iota 5814  df-recs 5887  df-en 6346  df-riota 6512  df-card 7098  df-ac 7267
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