Theorem zorn2g 10411

Description: Zorn's Lemma of [Monk1] p. 117. This version of zorn2 10414 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((𝑤 ⊆ 𝐴 ∧ 𝑅 Or 𝑤) → ∃𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝑤 (𝑧𝑅𝑥 ∨ 𝑧 = 𝑥))) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦)

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑅   𝑥,𝐴,𝑦,𝑧,𝑤

Proof of Theorem zorn2g

Dummy variables 𝑣 𝑢 𝑔 ℎ 𝑡 𝑠 𝑟 𝑞 𝑑 𝑘 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5099	. . . . . . . . 9 ⊢ (𝑔 = 𝑘 → (𝑔𝑞𝑛 ↔ 𝑘𝑞𝑛))
2	1	notbid 318	. . . . . . . 8 ⊢ (𝑔 = 𝑘 → (¬ 𝑔𝑞𝑛 ↔ ¬ 𝑘𝑞𝑛))
3	2	cbvralvw 3212	. . . . . . 7 ⊢ (∀𝑔 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑛)
4		breq2 5100	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑘𝑞𝑛 ↔ 𝑘𝑞𝑚))
5	4	notbid 318	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (¬ 𝑘𝑞𝑛 ↔ ¬ 𝑘𝑞𝑚))
6	5	ralbidv 3157	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
7	3, 6	bitrid 283	. . . . . 6 ⊢ (𝑛 = 𝑚 → (∀𝑔 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
8	7	cbvriotavw 7323	. . . . 5 ⊢ (℩𝑛 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛) = (℩𝑚 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚)
9		rneq 5883	. . . . . . . 8 ⊢ (ℎ = 𝑑 → ran ℎ = ran 𝑑)
10	9	raleqdv 3294	. . . . . . 7 ⊢ (ℎ = 𝑑 → (∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣 ↔ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣))
11	10	rabbidv 3404	. . . . . 6 ⊢ (ℎ = 𝑑 → {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} = {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣})
12	11	raleqdv 3294	. . . . . 6 ⊢ (ℎ = 𝑑 → (∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚 ↔ ∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
13	11, 12	riotaeqbidv 7316	. . . . 5 ⊢ (ℎ = 𝑑 → (℩𝑚 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚) = (℩𝑚 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
14	8, 13	eqtrid 2781	. . . 4 ⊢ (ℎ = 𝑑 → (℩𝑛 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛) = (℩𝑚 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
15	14	cbvmptv 5200	. . 3 ⊢ (ℎ ∈ V ↦ (℩𝑛 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛)) = (𝑑 ∈ V ↦ (℩𝑚 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
16		recseq 8303	. . 3 ⊢ ((ℎ ∈ V ↦ (℩𝑛 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛)) = (𝑑 ∈ V ↦ (℩𝑚 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚)) → recs((ℎ ∈ V ↦ (℩𝑛 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛))) = recs((𝑑 ∈ V ↦ (℩𝑚 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))))
17	15, 16	ax-mp 5	. 2 ⊢ recs((ℎ ∈ V ↦ (℩𝑛 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛))) = recs((𝑑 ∈ V ↦ (℩𝑚 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣}∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚)))
18		breq1 5099	. . . . 5 ⊢ (𝑞 = 𝑠 → (𝑞𝑅𝑣 ↔ 𝑠𝑅𝑣))
19	18	cbvralvw 3212	. . . 4 ⊢ (∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑣)
20		breq2 5100	. . . . 5 ⊢ (𝑣 = 𝑟 → (𝑠𝑅𝑣 ↔ 𝑠𝑅𝑟))
21	20	ralbidv 3157	. . . 4 ⊢ (𝑣 = 𝑟 → (∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟))
22	19, 21	bitrid 283	. . 3 ⊢ (𝑣 = 𝑟 → (∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟))
23	22	cbvrabv 3407	. 2 ⊢ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} = {𝑟 ∈ 𝐴 ∣ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟}
24		eqid 2734	. 2 ⊢ {𝑟 ∈ 𝐴 ∣ ∀𝑠 ∈ (recs((ℎ ∈ V ↦ (℩𝑛 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛))) “ 𝑢)𝑠𝑅𝑟} = {𝑟 ∈ 𝐴 ∣ ∀𝑠 ∈ (recs((ℎ ∈ V ↦ (℩𝑛 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛))) “ 𝑢)𝑠𝑅𝑟}
25		eqid 2734	. 2 ⊢ {𝑟 ∈ 𝐴 ∣ ∀𝑠 ∈ (recs((ℎ ∈ V ↦ (℩𝑛 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛))) “ 𝑡)𝑠𝑅𝑟} = {𝑟 ∈ 𝐴 ∣ ∀𝑠 ∈ (recs((ℎ ∈ V ↦ (℩𝑛 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣}∀𝑔 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛))) “ 𝑡)𝑠𝑅𝑟}
26	17, 23, 24, 25	zorn2lem7 10410	1 ⊢ ((𝐴 ∈ dom card ∧ 𝑅 Po 𝐴 ∧ ∀𝑤((𝑤 ⊆ 𝐴 ∧ 𝑅 Or 𝑤) → ∃𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝑤 (𝑧𝑅𝑥 ∨ 𝑧 = 𝑥))) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  {crab 3397  Vcvv 3438   ⊆ wss 3899   class class class wbr 5096   ↦ cmpt 5177   Po wpo 5528   Or wor 5529  dom cdm 5622  ran crn 5623   “ cima 5625  ℩crio 7312  recscrecs 8300  cardccrd 9845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-en 8882  df-card 9849

This theorem is referenced by:  zorng  10412  zorn2  10414