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Theorem zorn 7606
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 7605 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 7551 . . 3  |-  ( A  e.  _V  ->  A  e.  dom  card )
31, 2ax-mp 8 . 2  |-  A  e. 
dom  card
4 zorng 7603 . 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 643 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 1439    e. wcel 1520   A.wral 2272   E.wrex 2273   _Vcvv 2473    C_ wss 2791    C. wpss 2792   U.cuni 3422    Or wor 3862   dom cdm 4254   [ C_] crpss 5751   cardccrd 7046
This theorem is referenced by:  alexsubALTlem2  15892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1440  ax-6 1441  ax-7 1442  ax-gen 1443  ax-8 1522  ax-11 1523  ax-13 1524  ax-14 1525  ax-17 1527  ax-12o 1560  ax-10 1574  ax-9 1580  ax-4 1587  ax-16 1773  ax-ext 2044  ax-rep 3689  ax-sep 3699  ax-nul 3707  ax-pow 3743  ax-pr 3767  ax-un 4059  ax-ac 7544
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 895  df-3an 896  df-ex 1445  df-sb 1734  df-eu 1956  df-mo 1957  df-clab 2050  df-cleq 2055  df-clel 2058  df-ne 2182  df-ral 2276  df-rex 2277  df-reu 2278  df-rab 2279  df-v 2475  df-sbc 2649  df-csb 2731  df-dif 2794  df-un 2796  df-in 2798  df-ss 2802  df-pss 2804  df-nul 3071  df-if 3180  df-pw 3241  df-sn 3259  df-pr 3260  df-tp 3261  df-op 3262  df-uni 3423  df-int 3457  df-iun 3500  df-br 3585  df-opab 3639  df-mpt 3640  df-tr 3672  df-eprel 3854  df-id 3858  df-po 3863  df-so 3864  df-fr 3901  df-se 3902  df-we 3903  df-ord 3944  df-on 3945  df-suc 3947  df-xp 4268  df-rel 4269  df-cnv 4270  df-co 4271  df-dm 4272  df-rn 4273  df-res 4274  df-ima 4275  df-fun 4276  df-fn 4277  df-f 4278  df-f1 4279  df-fo 4280  df-f1o 4281  df-fv 4282  df-iso 4283  df-rpss 5752  df-iota 5766  df-recs 5839  df-en 6298  df-riota 6464  df-card 7050  df-ac 7219
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