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Theorem zorng 8015
 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 8018 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
StepHypRef Expression
1 risset 2552 . . . . . 6
2 eqimss2 3152 . . . . . . . . 9
3 unissb 3755 . . . . . . . . 9
42, 3sylib 190 . . . . . . . 8
5 vex 2730 . . . . . . . . . . . 12
65brrpss 6132 . . . . . . . . . . 11 []
76orbi1i 508 . . . . . . . . . 10 []
8 sspss 3195 . . . . . . . . . 10
97, 8bitr4i 245 . . . . . . . . 9 []
109ralbii 2531 . . . . . . . 8 []
114, 10sylibr 205 . . . . . . 7 []
1211reximi 2612 . . . . . 6 []
131, 12sylbi 189 . . . . 5 []
1413imim2i 15 . . . 4 [] [] []
1514alimi 1546 . . 3 [] [] []
16 porpss 6133 . . . 4 []
17 zorn2g 8014 . . . 4 [] [] [] []
1816, 17mp3an2 1270 . . 3 [] [] []
1915, 18sylan2 462 . 2 [] []
20 vex 2730 . . . . . 6
2120brrpss 6132 . . . . 5 []
2221notbii 289 . . . 4 []
2322ralbii 2531 . . 3 []
2423rexbii 2532 . 2 []
2519, 24sylib 190 1 []
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wo 359   wa 360  wal 1532   wceq 1619   wcel 1621  wral 2509  wrex 2510   wss 3078   wpss 3079  cuni 3727   class class class wbr 3920   wpo 4205   wor 4206   cdm 4580   [] crpss 6128  ccrd 7452 This theorem is referenced by:  zornn0g  8016  zorn  8018 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-rpss 6129  df-iota 6143  df-riota 6190  df-recs 6274  df-en 6750  df-card 7456
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