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Theorem zorng 8373
 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 8376 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 2745 . . . . . 6
2 eqimss2 3393 . . . . . . . . 9
3 unissb 4037 . . . . . . . . 9
42, 3sylib 189 . . . . . . . 8
5 vex 2951 . . . . . . . . . . . 12
65brrpss 6516 . . . . . . . . . . 11 []
76orbi1i 507 . . . . . . . . . 10 []
8 sspss 3438 . . . . . . . . . 10
97, 8bitr4i 244 . . . . . . . . 9 []
109ralbii 2721 . . . . . . . 8 []
114, 10sylibr 204 . . . . . . 7 []
1211reximi 2805 . . . . . 6 []
131, 12sylbi 188 . . . . 5 []
1413imim2i 14 . . . 4 [] [] []
1514alimi 1568 . . 3 [] [] []
16 porpss 6517 . . . 4 []
17 zorn2g 8372 . . . 4 [] [] [] []
1816, 17mp3an2 1267 . . 3 [] [] []
1915, 18sylan2 461 . 2 [] []
20 vex 2951 . . . . . 6
2120brrpss 6516 . . . . 5 []
2221notbii 288 . . . 4 []
2322ralbii 2721 . . 3 []
2423rexbii 2722 . 2 []
2519, 24sylib 189 1 []
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wo 358   wa 359  wal 1549   wceq 1652   wcel 1725  wral 2697  wrex 2698   wss 3312   wpss 3313  cuni 4007   class class class wbr 4204   wpo 4493   wor 4494   cdm 4869   [] crpss 6512  ccrd 7811 This theorem is referenced by:  zornn0g  8374  zorn  8376 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-rpss 6513  df-riota 6540  df-recs 6624  df-en 7101  df-card 7815
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