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Theorem zorn 7616
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 7615 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 7561 . . 3  |-  ( A  e.  _V  ->  A  e.  dom  card )
31, 2ax-mp 8 . 2  |-  A  e. 
dom  card
4 zorng 7613 . 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 644 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 1441    e. wcel 1522   A.wral 2275   E.wrex 2276   _Vcvv 2476    C_ wss 2794    C. wpss 2795   U.cuni 3430    Or wor 3870   dom cdm 4262   [ C_] crpss 5761   cardccrd 7056
This theorem is referenced by:  alexsubALTlem2  16446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1442  ax-6 1443  ax-7 1444  ax-gen 1445  ax-8 1524  ax-11 1525  ax-13 1526  ax-14 1527  ax-17 1529  ax-12o 1563  ax-10 1577  ax-9 1583  ax-4 1590  ax-16 1776  ax-ext 2047  ax-rep 3697  ax-sep 3707  ax-nul 3715  ax-pow 3751  ax-pr 3775  ax-un 4067  ax-ac 7554
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 897  df-3an 898  df-ex 1447  df-sb 1737  df-eu 1959  df-mo 1960  df-clab 2053  df-cleq 2058  df-clel 2061  df-ne 2185  df-ral 2279  df-rex 2280  df-reu 2281  df-rab 2282  df-v 2478  df-sbc 2652  df-csb 2734  df-dif 2797  df-un 2799  df-in 2801  df-ss 2805  df-pss 2807  df-nul 3074  df-if 3183  df-pw 3244  df-sn 3262  df-pr 3263  df-tp 3264  df-op 3265  df-uni 3431  df-int 3465  df-iun 3508  df-br 3593  df-opab 3647  df-mpt 3648  df-tr 3680  df-eprel 3862  df-id 3866  df-po 3871  df-so 3872  df-fr 3909  df-se 3910  df-we 3911  df-ord 3952  df-on 3953  df-suc 3955  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-res 4282  df-ima 4283  df-fun 4284  df-fn 4285  df-f 4286  df-f1 4287  df-fo 4288  df-f1o 4289  df-fv 4290  df-iso 4291  df-rpss 5762  df-iota 5776  df-recs 5849  df-en 6308  df-riota 6474  df-card 7060  df-ac 7229
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