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Theorem zorng 9278
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 9281 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 3056 . . . . . 6 ( 𝑧𝐴 ↔ ∃𝑥𝐴 𝑥 = 𝑧)
2 eqimss2 3642 . . . . . . . . 9 (𝑥 = 𝑧 𝑧𝑥)
3 unissb 4440 . . . . . . . . 9 ( 𝑧𝑥 ↔ ∀𝑢𝑧 𝑢𝑥)
42, 3sylib 208 . . . . . . . 8 (𝑥 = 𝑧 → ∀𝑢𝑧 𝑢𝑥)
5 vex 3192 . . . . . . . . . . . 12 𝑥 ∈ V
65brrpss 6900 . . . . . . . . . . 11 (𝑢 [] 𝑥𝑢𝑥)
76orbi1i 542 . . . . . . . . . 10 ((𝑢 [] 𝑥𝑢 = 𝑥) ↔ (𝑢𝑥𝑢 = 𝑥))
8 sspss 3689 . . . . . . . . . 10 (𝑢𝑥 ↔ (𝑢𝑥𝑢 = 𝑥))
97, 8bitr4i 267 . . . . . . . . 9 ((𝑢 [] 𝑥𝑢 = 𝑥) ↔ 𝑢𝑥)
109ralbii 2975 . . . . . . . 8 (∀𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥) ↔ ∀𝑢𝑧 𝑢𝑥)
114, 10sylibr 224 . . . . . . 7 (𝑥 = 𝑧 → ∀𝑢𝑧 (𝑢 [] 𝑥𝑢 = 𝑥))
1211reximi 3006 . . . . . 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 6901 . . . 4 [] Po 𝐴
17 zorn2g 9277 . . . 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 3192 . . . . . 6 𝑦 ∈ V
2120brrpss 6900 . . . . 5 (𝑥 [] 𝑦𝑥𝑦)
2221notbii 310 . . . 4 𝑥 [] 𝑦 ↔ ¬ 𝑥𝑦)
2322ralbii 2975 . . 3 (∀𝑦𝐴 ¬ 𝑥 [] 𝑦 ↔ ∀𝑦𝐴 ¬ 𝑥𝑦)
2423rexbii 3035 . 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 2907  wrex 2908  wss 3559  wpss 3560   cuni 4407   class class class wbr 4618   Po wpo 4998   Or wor 4999  dom cdm 5079   [] crpss 6896  cardccrd 8713
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-rpss 6897  df-wrecs 7359  df-recs 7420  df-en 7908  df-card 8717
This theorem is referenced by:  zornn0g  9279  zorn  9281
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