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Theorem zorn 8918
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 8917 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 8881 . . 3  |-  ( A  e.  _V  ->  A  e.  dom  card )
31, 2ax-mp 5 . 2  |-  A  e. 
dom  card
4 zorng 8915 . 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 668 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367   A.wal 1403    e. wcel 1842   A.wral 2753   E.wrex 2754   _Vcvv 3058    C_ wss 3413    C. wpss 3414   U.cuni 4190    Or wor 4742   dom cdm 4822   [ C.] crpss 6560   cardccrd 8347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-ac2 8874
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-rpss 6561  df-wrecs 7012  df-recs 7074  df-en 7554  df-card 8351  df-ac 8528
This theorem is referenced by:  alexsubALTlem2  20838
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