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Theorem zorn2g 9185
Description: Zorn's Lemma of [Monk1] p. 117. This version of zorn2 9188 avoids the Axiom of Choice by assuming that 𝐴 is well-orderable. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
Assertion
Ref Expression
zorn2g ((𝐴 ∈ dom card ∧ 𝑅 Po 𝐴 ∧ ∀𝑤((𝑤𝐴𝑅 Or 𝑤) → ∃𝑥𝐴𝑧𝑤 (𝑧𝑅𝑥𝑧 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦)
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑅   𝑥,𝐴,𝑦,𝑧,𝑤

Proof of Theorem zorn2g
Dummy variables 𝑣 𝑢 𝑔 𝑡 𝑠 𝑟 𝑞 𝑑 𝑘 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4580 . . . . . . . . 9 (𝑔 = 𝑘 → (𝑔𝑞𝑛𝑘𝑞𝑛))
21notbid 306 . . . . . . . 8 (𝑔 = 𝑘 → (¬ 𝑔𝑞𝑛 ↔ ¬ 𝑘𝑞𝑛))
32cbvralv 3146 . . . . . . 7 (∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑛)
4 breq2 4581 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑘𝑞𝑛𝑘𝑞𝑚))
54notbid 306 . . . . . . . 8 (𝑛 = 𝑚 → (¬ 𝑘𝑞𝑛 ↔ ¬ 𝑘𝑞𝑚))
65ralbidv 2968 . . . . . . 7 (𝑛 = 𝑚 → (∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
73, 6syl5bb 270 . . . . . 6 (𝑛 = 𝑚 → (∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
87cbvriotav 6500 . . . . 5 (𝑛 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛) = (𝑚 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚)
9 rneq 5259 . . . . . . . 8 ( = 𝑑 → ran = ran 𝑑)
109raleqdv 3120 . . . . . . 7 ( = 𝑑 → (∀𝑞 ∈ ran 𝑞𝑅𝑣 ↔ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣))
1110rabbidv 3163 . . . . . 6 ( = 𝑑 → {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} = {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣})
1211raleqdv 3120 . . . . . 6 ( = 𝑑 → (∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
1311, 12riotaeqbidv 6492 . . . . 5 ( = 𝑑 → (𝑚 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚) = (𝑚 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
148, 13syl5eq 2655 . . . 4 ( = 𝑑 → (𝑛 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛) = (𝑚 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
1514cbvmptv 4672 . . 3 ( ∈ V ↦ (𝑛 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛)) = (𝑑 ∈ V ↦ (𝑚 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
16 recseq 7334 . . 3 (( ∈ V ↦ (𝑛 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛)) = (𝑑 ∈ V ↦ (𝑚 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚)) → recs(( ∈ V ↦ (𝑛 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛))) = recs((𝑑 ∈ V ↦ (𝑚 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))))
1715, 16ax-mp 5 . 2 recs(( ∈ V ↦ (𝑛 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛))) = recs((𝑑 ∈ V ↦ (𝑚 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚)))
18 breq1 4580 . . . . 5 (𝑞 = 𝑠 → (𝑞𝑅𝑣𝑠𝑅𝑣))
1918cbvralv 3146 . . . 4 (∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑣)
20 breq2 4581 . . . . 5 (𝑣 = 𝑟 → (𝑠𝑅𝑣𝑠𝑅𝑟))
2120ralbidv 2968 . . . 4 (𝑣 = 𝑟 → (∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟))
2219, 21syl5bb 270 . . 3 (𝑣 = 𝑟 → (∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟))
2322cbvrabv 3171 . 2 {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} = {𝑟𝐴 ∣ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟}
24 eqid 2609 . 2 {𝑟𝐴 ∣ ∀𝑠 ∈ (recs(( ∈ V ↦ (𝑛 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛))) “ 𝑢)𝑠𝑅𝑟} = {𝑟𝐴 ∣ ∀𝑠 ∈ (recs(( ∈ V ↦ (𝑛 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛))) “ 𝑢)𝑠𝑅𝑟}
25 eqid 2609 . 2 {𝑟𝐴 ∣ ∀𝑠 ∈ (recs(( ∈ V ↦ (𝑛 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛))) “ 𝑡)𝑠𝑅𝑟} = {𝑟𝐴 ∣ ∀𝑠 ∈ (recs(( ∈ V ↦ (𝑛 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛))) “ 𝑡)𝑠𝑅𝑟}
2617, 23, 24, 25zorn2lem7 9184 1 ((𝐴 ∈ dom card ∧ 𝑅 Po 𝐴 ∧ ∀𝑤((𝑤𝐴𝑅 Or 𝑤) → ∃𝑥𝐴𝑧𝑤 (𝑧𝑅𝑥𝑧 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 381  wa 382  w3a 1030  wal 1472   = wceq 1474  wcel 1976  wral 2895  wrex 2896  {crab 2899  Vcvv 3172  wss 3539   class class class wbr 4577  cmpt 4637   Po wpo 4947   Or wor 4948  dom cdm 5028  ran crn 5029  cima 5031  crio 6488  recscrecs 7331  cardccrd 8621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-wrecs 7271  df-recs 7332  df-en 7819  df-card 8625
This theorem is referenced by:  zorng  9186  zorn2  9188
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