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Theorem zorn 7620
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 7619 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 7565 . . 3  |-  ( A  e.  _V  ->  A  e.  dom  card )
31, 2ax-mp 8 . 2  |-  A  e. 
dom  card
4 zorng 7617 . 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 1445    e. wcel 1526   A.wral 2279   E.wrex 2280   _Vcvv 2480    C_ wss 2798    C. wpss 2799   U.cuni 3434    Or wor 3874   dom cdm 4266   [ C.] crpss 5765   cardccrd 7060
This theorem is referenced by:  alexsubALTlem2  16450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1446  ax-6 1447  ax-7 1448  ax-gen 1449  ax-8 1528  ax-11 1529  ax-13 1530  ax-14 1531  ax-17 1533  ax-12o 1567  ax-10 1581  ax-9 1587  ax-4 1594  ax-16 1780  ax-ext 2051  ax-rep 3701  ax-sep 3711  ax-nul 3719  ax-pow 3755  ax-pr 3779  ax-un 4071  ax-ac 7558
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 897  df-3an 898  df-ex 1451  df-sb 1741  df-eu 1963  df-mo 1964  df-clab 2057  df-cleq 2062  df-clel 2065  df-ne 2189  df-ral 2283  df-rex 2284  df-reu 2285  df-rab 2286  df-v 2482  df-sbc 2656  df-csb 2738  df-dif 2801  df-un 2803  df-in 2805  df-ss 2809  df-pss 2811  df-nul 3078  df-if 3187  df-pw 3248  df-sn 3266  df-pr 3267  df-tp 3268  df-op 3269  df-uni 3435  df-int 3469  df-iun 3512  df-br 3597  df-opab 3651  df-mpt 3652  df-tr 3684  df-eprel 3866  df-id 3870  df-po 3875  df-so 3876  df-fr 3913  df-se 3914  df-we 3915  df-ord 3956  df-on 3957  df-suc 3959  df-xp 4280  df-rel 4281  df-cnv 4282  df-co 4283  df-dm 4284  df-rn 4285  df-res 4286  df-ima 4287  df-fun 4288  df-fn 4289  df-f 4290  df-f1 4291  df-fo 4292  df-f1o 4293  df-fv 4294  df-iso 4295  df-rpss 5766  df-iota 5780  df-recs 5853  df-en 6312  df-riota 6478  df-card 7064  df-ac 7233
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