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Theorem zorng 8923
 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. Theorem 6M of [Enderton] p. 151. This version of zorn 8926 avoids the Axiom of Choice by assuming that is well-orderable. (Contributed by NM, 12-Aug-2004.) (Revised by Mario Carneiro, 9-May-2015.)
Assertion
Ref Expression
zorng []
Distinct variable group:   ,,,

Proof of Theorem zorng
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 risset 2951 . . . . . 6
2 eqimss2 3514 . . . . . . . . 9
3 unissb 4244 . . . . . . . . 9
42, 3sylib 199 . . . . . . . 8
5 vex 3081 . . . . . . . . . . . 12
65brrpss 6579 . . . . . . . . . . 11 []
76orbi1i 522 . . . . . . . . . 10 []
8 sspss 3561 . . . . . . . . . 10
97, 8bitr4i 255 . . . . . . . . 9 []
109ralbii 2854 . . . . . . . 8 []
114, 10sylibr 215 . . . . . . 7 []
1211reximi 2891 . . . . . 6 []
131, 12sylbi 198 . . . . 5 []
1413imim2i 16 . . . 4 [] [] []
1514alimi 1680 . . 3 [] [] []
16 porpss 6580 . . . 4 []
17 zorn2g 8922 . . . 4 [] [] [] []
1816, 17mp3an2 1348 . . 3 [] [] []
1915, 18sylan2 476 . 2 [] []
20 vex 3081 . . . . . 6
2120brrpss 6579 . . . . 5 []
2221notbii 297 . . . 4 []
2322ralbii 2854 . . 3 []
2423rexbii 2925 . 2 []
2519, 24sylib 199 1 []
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wo 369   wa 370  wal 1435   wceq 1437   wcel 1867  wral 2773  wrex 2774   wss 3433   wpss 3434  cuni 4213   class class class wbr 4417   wpo 4764   wor 4765   cdm 4845   [] crpss 6575  ccrd 8359 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-rpss 6576  df-wrecs 7027  df-recs 7089  df-en 7569  df-card 8363 This theorem is referenced by:  zornn0g  8924  zorn  8926
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