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Theorem zorn2g 9363
Description: Zorn's Lemma of [Monk1] p. 117. This version of zorn2 9366 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 4688 . . . . . . . . 9 (𝑔 = 𝑘 → (𝑔𝑞𝑛𝑘𝑞𝑛))
21notbid 307 . . . . . . . 8 (𝑔 = 𝑘 → (¬ 𝑔𝑞𝑛 ↔ ¬ 𝑘𝑞𝑛))
32cbvralv 3201 . . . . . . 7 (∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑛)
4 breq2 4689 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑘𝑞𝑛𝑘𝑞𝑚))
54notbid 307 . . . . . . . 8 (𝑛 = 𝑚 → (¬ 𝑘𝑞𝑛 ↔ ¬ 𝑘𝑞𝑚))
65ralbidv 3015 . . . . . . 7 (𝑛 = 𝑚 → (∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
73, 6syl5bb 272 . . . . . 6 (𝑛 = 𝑚 → (∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
87cbvriotav 6662 . . . . 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 5383 . . . . . . . 8 ( = 𝑑 → ran = ran 𝑑)
109raleqdv 3174 . . . . . . 7 ( = 𝑑 → (∀𝑞 ∈ ran 𝑞𝑅𝑣 ↔ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣))
1110rabbidv 3220 . . . . . 6 ( = 𝑑 → {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} = {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣})
1211raleqdv 3174 . . . . . 6 ( = 𝑑 → (∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
1311, 12riotaeqbidv 6654 . . . . 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 2697 . . . 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 4783 . . 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 7515 . . 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 4688 . . . . 5 (𝑞 = 𝑠 → (𝑞𝑅𝑣𝑠𝑅𝑣))
1918cbvralv 3201 . . . 4 (∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑣)
20 breq2 4689 . . . . 5 (𝑣 = 𝑟 → (𝑠𝑅𝑣𝑠𝑅𝑟))
2120ralbidv 3015 . . . 4 (𝑣 = 𝑟 → (∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟))
2219, 21syl5bb 272 . . 3 (𝑣 = 𝑟 → (∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟))
2322cbvrabv 3230 . 2 {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} = {𝑟𝐴 ∣ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟}
24 eqid 2651 . 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 2651 . 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 9362 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 382  wa 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607   class class class wbr 4685  cmpt 4762   Po wpo 5062   Or wor 5063  dom cdm 5143  ran crn 5144  cima 5146  crio 6650  recscrecs 7512  cardccrd 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-wrecs 7452  df-recs 7513  df-en 7998  df-card 8803
This theorem is referenced by:  zorng  9364  zorn2  9366
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