Theorem zorn2g 10522

Description: Zorn's Lemma of [Monk1] p. 117. This version of zorn2 10525 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 5127	. . . . . . . . 9 ⊢ (𝑔 = 𝑘 → (𝑔𝑞𝑛 ↔ 𝑘𝑞𝑛))
2	1	notbid 318	. . . . . . . 8 ⊢ (𝑔 = 𝑘 → (¬ 𝑔𝑞𝑛 ↔ ¬ 𝑘𝑞𝑛))
3	2	cbvralvw 3224	. . . . . . 7 ⊢ (∀𝑔 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑛)
4		breq2 5128	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑘𝑞𝑛 ↔ 𝑘𝑞𝑚))
5	4	notbid 318	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (¬ 𝑘𝑞𝑛 ↔ ¬ 𝑘𝑞𝑚))
6	5	ralbidv 3164	. . . . . . 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 7377	. . . . 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 5921	. . . . . . . 8 ⊢ (ℎ = 𝑑 → ran ℎ = ran 𝑑)
10	9	raleqdv 3309	. . . . . . 7 ⊢ (ℎ = 𝑑 → (∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣 ↔ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣))
11	10	rabbidv 3428	. . . . . 6 ⊢ (ℎ = 𝑑 → {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} = {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣})
12	11	raleqdv 3309	. . . . . 6 ⊢ (ℎ = 𝑑 → (∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚 ↔ ∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
13	11, 12	riotaeqbidv 7370	. . . . 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 2783	. . . 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 5230	. . 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 8393	. . 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 5127	. . . . 5 ⊢ (𝑞 = 𝑠 → (𝑞𝑅𝑣 ↔ 𝑠𝑅𝑣))
19	18	cbvralvw 3224	. . . 4 ⊢ (∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑣)
20		breq2 5128	. . . . 5 ⊢ (𝑣 = 𝑟 → (𝑠𝑅𝑣 ↔ 𝑠𝑅𝑟))
21	20	ralbidv 3164	. . . 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 3431	. 2 ⊢ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} = {𝑟 ∈ 𝐴 ∣ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟}
24		eqid 2736	. 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 2736	. 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 10521	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 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061  {crab 3420  Vcvv 3464   ⊆ wss 3931   class class class wbr 5124   ↦ cmpt 5206   Po wpo 5564   Or wor 5565  dom cdm 5659  ran crn 5660   “ cima 5662  ℩crio 7366  recscrecs 8389  cardccrd 9954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-en 8965  df-card 9958

This theorem is referenced by:  zorng  10523  zorn2  10525