Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  zorn Unicode version

Theorem zorn 8813
 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 8812 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.)
Hypothesis
Ref Expression
zornn0.1
Assertion
Ref Expression
zorn
Distinct variable group:   ,,,

Proof of Theorem zorn
StepHypRef Expression
1 zornn0.1 . . 3
2 numth3 8776 . . 3
31, 2ax-mp 5 . 2
4 zorng 8810 . 2
53, 4mpan 670 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  /\wa 369  A.wal 1368  e.wcel 1758  A.wral 2800  E.wrex 2801   cvv 3081  C_wss 3442  C.wpss 3443  U.cuni 4208  Orwor 4757  domcdm 4957   crpss 6492   ccrd 8242 This theorem is referenced by:  alexsubALTlem2  20019 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4520  ax-sep 4530  ax-nul 4538  ax-pow 4587  ax-pr 4648  ax-un 6505  ax-ac2 8769 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3083  df-sbc 3298  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3752  df-if 3906  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4209  df-int 4246  df-iun 4290  df-br 4410  df-opab 4468  df-mpt 4469  df-tr 4503  df-eprel 4749  df-id 4753  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-ord 4839  df-on 4840  df-suc 4842  df-xp 4963  df-rel 4964  df-cnv 4965  df-co 4966  df-dm 4967  df-rn 4968  df-res 4969  df-ima 4970  df-iota 5500  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-isom 5546  df-riota 6183  df-rpss 6493  df-recs 6966  df-en 7445  df-card 8246  df-ac 8423
 Copyright terms: Public domain W3C validator