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Theorem zorn 8150
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 8149 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 8113 . . 3  |-  ( A  e.  _V  ->  A  e.  dom  card )
31, 2ax-mp 8 . 2  |-  A  e. 
dom  card
4 zorng 8147 . 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 651 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 358   A.wal 1530    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165    C. wpss 3166   U.cuni 3843    Or wor 4329   dom cdm 4705   [ C.] crpss 6292   cardccrd 7584
This theorem is referenced by:  alexsubALTlem2  17758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-ac2 8105
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-rpss 6293  df-riota 6320  df-recs 6404  df-en 6880  df-card 7588  df-ac 7759
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