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Theorem zorn2g 9526
 Description: Zorn's Lemma of [Monk1] p. 117. This version of zorn2 9529 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 4787 . . . . . . . . 9 (𝑔 = 𝑘 → (𝑔𝑞𝑛𝑘𝑞𝑛))
21notbid 307 . . . . . . . 8 (𝑔 = 𝑘 → (¬ 𝑔𝑞𝑛 ↔ ¬ 𝑘𝑞𝑛))
32cbvralv 3319 . . . . . . 7 (∀𝑔 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑛)
4 breq2 4788 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑘𝑞𝑛𝑘𝑞𝑚))
54notbid 307 . . . . . . . 8 (𝑛 = 𝑚 → (¬ 𝑘𝑞𝑛 ↔ ¬ 𝑘𝑞𝑚))
65ralbidv 3134 . . . . . . 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 6764 . . . . 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 5489 . . . . . . . 8 ( = 𝑑 → ran = ran 𝑑)
109raleqdv 3292 . . . . . . 7 ( = 𝑑 → (∀𝑞 ∈ ran 𝑞𝑅𝑣 ↔ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣))
1110rabbidv 3338 . . . . . 6 ( = 𝑑 → {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} = {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣})
1211raleqdv 3292 . . . . . 6 ( = 𝑑 → (∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚 ↔ ∀𝑘 ∈ {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
1311, 12riotaeqbidv 6756 . . . . 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 2816 . . . 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 4882 . . 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 7622 . . 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 4787 . . . . 5 (𝑞 = 𝑠 → (𝑞𝑅𝑣𝑠𝑅𝑣))
1918cbvralv 3319 . . . 4 (∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑣)
20 breq2 4788 . . . . 5 (𝑣 = 𝑟 → (𝑠𝑅𝑣𝑠𝑅𝑟))
2120ralbidv 3134 . . . 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 3348 . 2 {𝑣𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} = {𝑟𝐴 ∣ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟}
24 eqid 2770 . 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 2770 . 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 9525 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 382   ∨ wo 826   ∧ w3a 1070  ∀wal 1628   = wceq 1630   ∈ wcel 2144  ∀wral 3060  ∃wrex 3061  {crab 3064  Vcvv 3349   ⊆ wss 3721   class class class wbr 4784   ↦ cmpt 4861   Po wpo 5168   Or wor 5169  dom cdm 5249  ran crn 5250   “ cima 5252  ℩crio 6752  recscrecs 7619  cardccrd 8960 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-wrecs 7558  df-recs 7620  df-en 8109  df-card 8964 This theorem is referenced by:  zorng  9527  zorn2  9529
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