Theorem zorn 3611

Description: Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set A (with an ordering relation R) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zornlem1 3603 through zornlem7 3609; this final piece mainly changes bound variables to eliminate the hypotheses of zornlem7 3609.

Hypothesis

Ref	Expression
zorn.1	⊢ A ∈ V

Assertion

Ref	Expression
zorn	⊢ ((R Po A ∧ ∀w((w ⊆ A ∧ R Or w) → ∃x ∈ A ∀z ∈ w (zRx ∨ z = x))) → ∃x ∈ A ∀y ∈ A ¬ xRy)

Distinct variable group(s):   x,y,z,w,R   x,A,y,z,w

Proof of Theorem zorn

Step	Hyp	Ref	Expression
1		zorn.1	. 2 ⊢ A ∈ V
2		rdglem1 2975	. 2 ⊢ {a∣∃b ∈ On (a Fn b ∧ ∀c ∈ b (a ‘c) = ({⟨h, k⟩∣k = ∪{m ∈ {v ∈ A∣∀q ∈ ran hqRv}∣∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm}} ‘(a ↾ c)))} = {d∣∃f ∈ On (d Fn f ∧ ∀g ∈ f (d ‘g) = ({⟨h, k⟩∣k = ∪{m ∈ {v ∈ A∣∀q ∈ ran hqRv}∣∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm}} ‘(d ↾ g)))}
3		cleqid 1102	. 2 ⊢ ∪{a∣∃b ∈ On (a Fn b ∧ ∀c ∈ b (a ‘c) = ({⟨h, k⟩∣k = ∪{m ∈ {v ∈ A∣∀q ∈ ran hqRv}∣∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm}} ‘(a ↾ c)))} = ∪{a∣∃b ∈ On (a Fn b ∧ ∀c ∈ b (a ‘c) = ({⟨h, k⟩∣k = ∪{m ∈ {v ∈ A∣∀q ∈ ran hqRv}∣∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm}} ‘(a ↾ c)))}
4		breq2 2066	. . . . 5 ⊢ (v = r → (sRv ↔ sRr))
5	4	biraldv 1219	. . . 4 ⊢ (v = r → (∀s ∈ ran dsRv ↔ ∀s ∈ ran dsRr))
6		breq1 2065	. . . . 5 ⊢ (q = s → (qRv ↔ sRv))
7	6	cbvralv 1333	. . . 4 ⊢ (∀q ∈ ran dqRv ↔ ∀s ∈ ran dsRv)
8	5, 7	syl5bb 410	. . 3 ⊢ (v = r → (∀q ∈ ran dqRv ↔ ∀s ∈ ran dsRr))
9	8	cbvrabv 1426	. 2 ⊢ {v ∈ A∣∀q ∈ ran dqRv} = {r ∈ A∣∀s ∈ ran dsRr}
10		cleqid 1102	. 2 ⊢ {r ∈ A∣∀s ∈ (∪{a∣∃b ∈ On (a Fn b ∧ ∀c ∈ b (a ‘c) = ({⟨h, k⟩∣k = ∪{m ∈ {v ∈ A∣∀q ∈ ran hqRv}∣∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm}} ‘(a ↾ c)))} “ t)sRr} = {r ∈ A∣∀s ∈ (∪{a∣∃b ∈ On (a Fn b ∧ ∀c ∈ b (a ‘c) = ({⟨h, k⟩∣k = ∪{m ∈ {v ∈ A∣∀q ∈ ran hqRv}∣∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm}} ‘(a ↾ c)))} “ t)sRr}
11		id 9	. . . 4 ⊢ (k = g → k = g)
12		rneq 2555	. . . . . . . . . . . 12 ⊢ (h = d → ran h = ran d)
13		raleq 1324	. . . . . . . . . . . 12 ⊢ (ran h = ran d → (∀q ∈ ran hqRv ↔ ∀q ∈ ran dqRv))
14	12, 13	syl 12	. . . . . . . . . . 11 ⊢ (h = d → (∀q ∈ ran hqRv ↔ ∀q ∈ ran dqRv))
15	14	birabsdv 1344	. . . . . . . . . 10 ⊢ (h = d → {v ∈ A∣∀q ∈ ran hqRv} = {v ∈ A∣∀q ∈ ran dqRv})
16	15	eleq2d 1156	. . . . . . . . 9 ⊢ (h = d → (n ∈ {v ∈ A∣∀q ∈ ran hqRv} ↔ n ∈ {v ∈ A∣∀q ∈ ran dqRv}))
17		raleq 1324	. . . . . . . . . . 11 ⊢ ({v ∈ A∣∀q ∈ ran hqRv} = {v ∈ A∣∀q ∈ ran dqRv} → (∀j ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ jqn ↔ ∀j ∈ {v ∈ A∣∀q ∈ ran dqRv} ¬ jqn))
18		breq1 2065	. . . . . . . . . . . . 13 ⊢ (k = j → (kqn ↔ jqn))
19	18	negbid 463	. . . . . . . . . . . 12 ⊢ (k = j → (¬ kqn ↔ ¬ jqn))
20	19	cbvralv 1333	. . . . . . . . . . 11 ⊢ (∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqn ↔ ∀j ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ jqn)
21	17, 20	syl5bb 410	. . . . . . . . . 10 ⊢ ({v ∈ A∣∀q ∈ ran hqRv} = {v ∈ A∣∀q ∈ ran dqRv} → (∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqn ↔ ∀j ∈ {v ∈ A∣∀q ∈ ran dqRv} ¬ jqn))
22	15, 21	syl 12	. . . . . . . . 9 ⊢ (h = d → (∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqn ↔ ∀j ∈ {v ∈ A∣∀q ∈ ran dqRv} ¬ jqn))
23	16, 22	anbi12d 476	. . . . . . . 8 ⊢ (h = d → ((n ∈ {v ∈ A∣∀q ∈ ran hqRv} ∧ ∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqn) ↔ (n ∈ {v ∈ A∣∀q ∈ ran dqRv} ∧ ∀j ∈ {v ∈ A∣∀q ∈ ran dqRv} ¬ jqn)))
24	23	biabdv 1183	. . . . . . 7 ⊢ (h = d → {n∣(n ∈ {v ∈ A∣∀q ∈ ran hqRv} ∧ ∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqn)} = {n∣(n ∈ {v ∈ A∣∀q ∈ ran dqRv} ∧ ∀j ∈ {v ∈ A∣∀q ∈ ran dqRv} ¬ jqn)})
25		eleq1 1149	. . . . . . . . 9 ⊢ (m = n → (m ∈ {v ∈ A∣∀q ∈ ran hqRv} ↔ n ∈ {v ∈ A∣∀q ∈ ran hqRv}))
26		breq2 2066	. . . . . . . . . . 11 ⊢ (m = n → (kqm ↔ kqn))
27	26	negbid 463	. . . . . . . . . 10 ⊢ (m = n → (¬ kqm ↔ ¬ kqn))
28	27	biraldv 1219	. . . . . . . . 9 ⊢ (m = n → (∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm ↔ ∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqn))
29	25, 28	anbi12d 476	. . . . . . . 8 ⊢ (m = n → ((m ∈ {v ∈ A∣∀q ∈ ran hqRv} ∧ ∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm) ↔ (n ∈ {v ∈ A∣∀q ∈ ran hqRv} ∧ ∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqn)))
30	29	cbvabv 1424	. . . . . . 7 ⊢ {m∣(m ∈ {v ∈ A∣∀q ∈ ran hqRv} ∧ ∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm)} = {n∣(n ∈ {v ∈ A∣∀q ∈ ran hqRv} ∧ ∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqn)}
31	24, 30	syl5eq 1136	. . . . . 6 ⊢ (h = d → {m∣(m ∈ {v ∈ A∣∀q ∈ ran hqRv} ∧ ∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm)} = {n∣(n ∈ {v ∈ A∣∀q ∈ ran dqRv} ∧ ∀j ∈ {v ∈ A∣∀q ∈ ran dqRv} ¬ jqn)})
32		df-rab 1208	. . . . . 6 ⊢ {m ∈ {v ∈ A∣∀q ∈ ran hqRv}∣∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm} = {m∣(m ∈ {v ∈ A∣∀q ∈ ran hqRv} ∧ ∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm)}
33		df-rab 1208	. . . . . 6 ⊢ {n ∈ {v ∈ A∣∀q ∈ ran dqRv}∣∀j ∈ {v ∈ A∣∀q ∈ ran dqRv} ¬ jqn} = {n∣(n ∈ {v ∈ A∣∀q ∈ ran dqRv} ∧ ∀j ∈ {v ∈ A∣∀q ∈ ran dqRv} ¬ jqn)}
34	31, 32, 33	3eqtr4g 1147	. . . . 5 ⊢ (h = d → {m ∈ {v ∈ A∣∀q ∈ ran hqRv}∣∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm} = {n ∈ {v ∈ A∣∀q ∈ ran dqRv}∣∀j ∈ {v ∈ A∣∀q ∈ ran dqRv} ¬ jqn})
35	34	unieqd 1929	. . . 4 ⊢ (h = d → ∪{m ∈ {v ∈ A∣∀q ∈ ran hqRv}∣∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm} = ∪{n ∈ {v ∈ A∣∀q ∈ ran dqRv}∣∀j ∈ {v ∈ A∣∀q ∈ ran dqRv} ¬ jqn})
36	11, 35	cleqan12rd 1117	. . 3 ⊢ ((h = d ∧ k = g) → (k = ∪{m ∈ {v ∈ A∣∀q ∈ ran hqRv}∣∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm} ↔ g = ∪{n ∈ {v ∈ A∣∀q ∈ ran dqRv}∣∀j ∈ {v ∈ A∣∀q ∈ ran dqRv} ¬ jqn}))
37	36	cbvopabv 2105	. 2 ⊢ {⟨h, k⟩∣k = ∪{m ∈ {v ∈ A∣∀q ∈ ran hqRv}∣∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm}} = {⟨d, g⟩∣g = ∪{n ∈ {v ∈ A∣∀q ∈ ran dqRv}∣∀j ∈ {v ∈ A∣∀q ∈ ran dqRv} ¬ jqn}}
38		cleqid 1102	. 2 ⊢ {r ∈ A∣∀s ∈ (∪{a∣∃b ∈ On (a Fn b ∧ ∀c ∈ b (a ‘c) = ({⟨h, k⟩∣k = ∪{m ∈ {v ∈ A∣∀q ∈ ran hqRv}∣∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm}} ‘(a ↾ c)))} “ u)sRr} = {r ∈ A∣∀s ∈ (∪{a∣∃b ∈ On (a Fn b ∧ ∀c ∈ b (a ‘c) = ({⟨h, k⟩∣k = ∪{m ∈ {v ∈ A∣∀q ∈ ran hqRv}∣∀k ∈ {v ∈ A∣∀q ∈ ran hqRv} ¬ kqm}} ‘(a ↾ c)))} “ u)sRr}
39	1, 2, 3, 9, 10, 37, 38	zornlem7 3609	1 ⊢ ((R Po A ∧ ∀w((w ⊆ A ∧ R Or w) → ∃x ∈ A ∀z ∈ w (zRx ∨ z = x))) → ∃x ∈ A ∀y ∈ A ¬ xRy)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196  ∀wal 672   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  {crab 1204  Vcvv 1348   ⊆ wss 1487  ∪cuni 1919   class class class wbr 2054  {copab 2055   Po wpo 2058   Or wor 2059  Oncon0 2199  ran crn 2411   ↾ cres 2412   “ cima 2413   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  zorn2 3612

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-iso 2439

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