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Theorem zorn 7360
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 7359 for a version with general partial orderings.
Hypothesis
Ref Expression
zornn0.1
Assertion
Ref Expression
zorn []
Distinct variable group:   ,,,

Proof of Theorem zorn
StepHypRef Expression
1 zornn0.1 . . 3
2 numth3 7344 . . 3
31, 2ax-mp 8 . 2
4 zorng 7357 . 2 []
53, 4mpan 646 1 []
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 357  wal 1442   wcel 1528  wral 2239  wrex 2240  cvv 2440   wss 2750   wpss 2751  cuni 3374   wor 3791   cdm 4172   [] crpss 5632  ccrd 6821
This theorem is referenced by:  alexsublem2 13569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1443  ax-6 1444  ax-7 1445  ax-gen 1446  ax-8 1530  ax-10 1531  ax-11 1532  ax-12 1533  ax-13 1534  ax-14 1535  ax-17 1542  ax-9 1557  ax-4 1563  ax-16 1741  ax-ext 2012  ax-rep 3616  ax-sep 3626  ax-nul 3635  ax-pow 3671  ax-pr 3695  ax-un 3973  ax-ac 7298
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-3or 900  df-3an 901  df-ex 1448  df-sb 1703  df-eu 1930  df-mo 1931  df-clab 2018  df-cleq 2023  df-clel 2026  df-ne 2149  df-ral 2243  df-rex 2244  df-reu 2245  df-rab 2246  df-v 2442  df-sbc 2609  df-csb 2691  df-dif 2753  df-un 2755  df-in 2757  df-ss 2761  df-pss 2763  df-nul 3026  df-if 3135  df-pw 3196  df-sn 3214  df-pr 3215  df-tp 3216  df-op 3217  df-uni 3375  df-int 3409  df-iun 3451  df-br 3532  df-opab 3585  df-tr 3600  df-eprel 3783  df-id 3787  df-po 3792  df-so 3806  df-fr 3826  df-we 3842  df-ord 3858  df-on 3859  df-suc 3861  df-xp 4186  df-rel 4187  df-cnv 4188  df-co 4189  df-dm 4190  df-rn 4191  df-res 4192  df-ima 4193  df-fun 4194  df-fn 4195  df-f 4196  df-f1 4197  df-fo 4198  df-f1o 4199  df-fv 4200  df-iso 4201  df-mpt 5371  df-rpss 5633  df-recs 5722  df-en 6121  df-card 6824  df-ac 6974
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