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Theorem zorn2g 10440
Description: Zorn's Lemma of [Monk1] p. 117. This version of zorn2 10443 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 5109 . . . . . . . . 9 (𝑔 = 𝑘 → (𝑔𝑞𝑛𝑘𝑞𝑛))
21notbid 318 . . . . . . . 8 (𝑔 = 𝑘 → (¬ 𝑔𝑞𝑛 ↔ ¬ 𝑘𝑞𝑛))
32cbvralvw 3226 . . . . . . 7 (∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑛)
4 breq2 5110 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑘𝑞𝑛𝑘𝑞𝑚))
54notbid 318 . . . . . . . 8 (𝑛 = 𝑚 → (¬ 𝑘𝑞𝑛 ↔ ¬ 𝑘𝑞𝑚))
65ralbidv 3175 . . . . . . 7 (𝑛 = 𝑚 → (∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
73, 6bitrid 283 . . . . . 6 (𝑛 = 𝑚 → (∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
87cbvriotavw 7324 . . . . 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 5892 . . . . . . . 8 ( = 𝑑 → ran = ran 𝑑)
109raleqdv 3314 . . . . . . 7 ( = 𝑑 → (∀𝑞 ∈ ran 𝑞𝑅𝑣 ↔ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣))
1110rabbidv 3416 . . . . . 6 ( = 𝑑 → {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} = {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣})
1211raleqdv 3314 . . . . . 6 ( = 𝑑 → (∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
1311, 12riotaeqbidv 7317 . . . . 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, 13eqtrid 2789 . . . 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 5219 . . 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 8321 . . 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 5109 . . . . 5 (𝑞 = 𝑠 → (𝑞𝑅𝑣𝑠𝑅𝑣))
1918cbvralvw 3226 . . . 4 (∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑣)
20 breq2 5110 . . . . 5 (𝑣 = 𝑟 → (𝑠𝑅𝑣𝑠𝑅𝑟))
2120ralbidv 3175 . . . 4 (𝑣 = 𝑟 → (∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟))
2219, 21bitrid 283 . . 3 (𝑣 = 𝑟 → (∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟))
2322cbvrabv 3418 . 2 {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} = {𝑟𝐴 ∣ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟}
24 eqid 2737 . 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 2737 . 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 10439 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 397  wo 846  w3a 1088  wal 1540   = wceq 1542  wcel 2107  wral 3065  wrex 3074  {crab 3408  Vcvv 3446  wss 3911   class class class wbr 5106  cmpt 5189   Po wpo 5544   Or wor 5545  dom cdm 5634  ran crn 5635  cima 5637  crio 7313  recscrecs 8317  cardccrd 9872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-en 8885  df-card 9876
This theorem is referenced by:  zorng  10441  zorn2  10443
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