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Theorem zorng 9286
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 9289 avoids the Axiom of Choice by assuming that 𝐴 is well-orderable. (Contributed by NM, 12-Aug-2004.) (Revised by Mario Carneiro, 9-May-2015.)
Assertion
Ref Expression
zorng ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem zorng
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 risset 3057 . . . . . 6 ( 𝑧𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑧)
2 eqimss2 3643 . . . . . . . . 9 (𝑥 = 𝑧 𝑧𝑥)
3 unissb 4442 . . . . . . . . 9 ( 𝑧𝑥 ↔ ∀𝑢𝑧 𝑢𝑥)
42, 3sylib 208 . . . . . . . 8 (𝑥 = 𝑧 → ∀𝑢𝑧 𝑢𝑥)
5 vex 3193 . . . . . . . . . . . 12 𝑥 ∈ V
65brrpss 6905 . . . . . . . . . . 11 (𝑢 [] 𝑥𝑢𝑥)
76orbi1i 542 . . . . . . . . . 10 ((𝑢 [] 𝑥𝑢 = 𝑥) ↔ (𝑢𝑥𝑢 = 𝑥))
8 sspss 3690 . . . . . . . . . 10 (𝑢𝑥 ↔ (𝑢𝑥𝑢 = 𝑥))
97, 8bitr4i 267 . . . . . . . . 9 ((𝑢 [] 𝑥𝑢 = 𝑥) ↔ 𝑢𝑥)
109ralbii 2976 . . . . . . . 8 (∀𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥) ↔ ∀𝑢𝑧 𝑢𝑥)
114, 10sylibr 224 . . . . . . 7 (𝑥 = 𝑧 → ∀𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))
1211reximi 3007 . . . . . 6 (∃𝑥𝐴 𝑥 = 𝑧 → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))
131, 12sylbi 207 . . . . 5 ( 𝑧𝐴 → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))
1413imim2i 16 . . . 4 (((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴) → ((𝑧𝐴 ∧ [] Or 𝑧) → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥)))
1514alimi 1736 . . 3 (∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴) → ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥)))
16 porpss 6906 . . . 4 [] Po 𝐴
17 zorn2g 9285 . . . 4 ((𝐴 ∈ dom card ∧ [] Po 𝐴 ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 [] 𝑦)
1816, 17mp3an2 1409 . . 3 ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → ∃𝑥𝐴𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 [] 𝑦)
1915, 18sylan2 491 . 2 ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 [] 𝑦)
20 vex 3193 . . . . . 6 𝑦 ∈ V
2120brrpss 6905 . . . . 5 (𝑥 [] 𝑦𝑥𝑦)
2221notbii 310 . . . 4 𝑥 [] 𝑦 ↔ ¬ 𝑥𝑦)
2322ralbii 2976 . . 3 (∀𝑦𝐴 ¬ 𝑥 [] 𝑦 ↔ ∀𝑦𝐴 ¬ 𝑥𝑦)
2423rexbii 3036 . 2 (∃𝑥𝐴𝑦𝐴 ¬ 𝑥 [] 𝑦 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
2519, 24sylib 208 1 ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384  wal 1478   = wceq 1480  wcel 1987  wral 2908  wrex 2909  wss 3560  wpss 3561   cuni 4409   class class class wbr 4623   Po wpo 5003   Or wor 5004  dom cdm 5084   [] crpss 6901  cardccrd 8721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-rpss 6902  df-wrecs 7367  df-recs 7428  df-en 7916  df-card 8725
This theorem is referenced by:  zornn0g  9287  zorn  9289
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