Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  zorn2g Structured version   Visualization version   GIF version

Theorem zorn2g 9963
 Description: Zorn's Lemma of [Monk1] p. 117. This version of zorn2 9966 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 5035 . . . . . . . . 9 (𝑔 = 𝑘 → (𝑔𝑞𝑛𝑘𝑞𝑛))
21notbid 321 . . . . . . . 8 (𝑔 = 𝑘 → (¬ 𝑔𝑞𝑛 ↔ ¬ 𝑘𝑞𝑛))
32cbvralvw 3361 . . . . . . 7 (∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑛)
4 breq2 5036 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑘𝑞𝑛𝑘𝑞𝑚))
54notbid 321 . . . . . . . 8 (𝑛 = 𝑚 → (¬ 𝑘𝑞𝑛 ↔ ¬ 𝑘𝑞𝑚))
65ralbidv 3126 . . . . . . 7 (𝑛 = 𝑚 → (∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
73, 6syl5bb 286 . . . . . 6 (𝑛 = 𝑚 → (∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
87cbvriotavw 7118 . . . . 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 5777 . . . . . . . 8 ( = 𝑑 → ran = ran 𝑑)
109raleqdv 3329 . . . . . . 7 ( = 𝑑 → (∀𝑞 ∈ ran 𝑞𝑅𝑣 ↔ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣))
1110rabbidv 3392 . . . . . 6 ( = 𝑑 → {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} = {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣})
1211raleqdv 3329 . . . . . 6 ( = 𝑑 → (∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
1311, 12riotaeqbidv 7111 . . . . 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 2805 . . . 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 5135 . . 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 8020 . . 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 5035 . . . . 5 (𝑞 = 𝑠 → (𝑞𝑅𝑣𝑠𝑅𝑣))
1918cbvralvw 3361 . . . 4 (∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑣)
20 breq2 5036 . . . . 5 (𝑣 = 𝑟 → (𝑠𝑅𝑣𝑠𝑅𝑟))
2120ralbidv 3126 . . . 4 (𝑣 = 𝑟 → (∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟))
2219, 21syl5bb 286 . . 3 (𝑣 = 𝑟 → (∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟))
2322cbvrabv 3404 . 2 {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} = {𝑟𝐴 ∣ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟}
24 eqid 2758 . 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 2758 . 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 9962 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   ∧ wa 399   ∨ wo 844   ∧ w3a 1084  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409   ⊆ wss 3858   class class class wbr 5032   ↦ cmpt 5112   Po wpo 5441   Or wor 5442  dom cdm 5524  ran crn 5525   “ cima 5527  ℩crio 7107  recscrecs 8017  cardccrd 9397 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-se 5484  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-isom 6344  df-riota 7108  df-wrecs 7957  df-recs 8018  df-en 8528  df-card 9401 This theorem is referenced by:  zorng  9964  zorn2  9966
 Copyright terms: Public domain W3C validator