Theorem zorn2g 10425

Description: Zorn's Lemma of [Monk1] p. 117. This version of zorn2 10428 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 5103	. . . . . . . . 9 ⊢ (𝑔 = 𝑘 → (𝑔𝑞𝑛 ↔ 𝑘𝑞𝑛))
2	1	notbid 318	. . . . . . . 8 ⊢ (𝑔 = 𝑘 → (¬ 𝑔𝑞𝑛 ↔ ¬ 𝑘𝑞𝑛))
3	2	cbvralvw 3216	. . . . . . 7 ⊢ (∀𝑔 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑔𝑞𝑛 ↔ ∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑛)
4		breq2 5104	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑘𝑞𝑛 ↔ 𝑘𝑞𝑚))
5	4	notbid 318	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (¬ 𝑘𝑞𝑛 ↔ ¬ 𝑘𝑞𝑚))
6	5	ralbidv 3161	. . . . . . 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 7335	. . . . 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 5893	. . . . . . . 8 ⊢ (ℎ = 𝑑 → ran ℎ = ran 𝑑)
10	9	raleqdv 3298	. . . . . . 7 ⊢ (ℎ = 𝑑 → (∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣 ↔ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣))
11	10	rabbidv 3408	. . . . . 6 ⊢ (ℎ = 𝑑 → {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} = {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣})
12	11	raleqdv 3298	. . . . . 6 ⊢ (ℎ = 𝑑 → (∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran ℎ 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚 ↔ ∀𝑘 ∈ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} ¬ 𝑘𝑞𝑚))
13	11, 12	riotaeqbidv 7328	. . . . 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 2784	. . . 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 5204	. . 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 8315	. . 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 5103	. . . . 5 ⊢ (𝑞 = 𝑠 → (𝑞𝑅𝑣 ↔ 𝑠𝑅𝑣))
19	18	cbvralvw 3216	. . . 4 ⊢ (∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣 ↔ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑣)
20		breq2 5104	. . . . 5 ⊢ (𝑣 = 𝑟 → (𝑠𝑅𝑣 ↔ 𝑠𝑅𝑟))
21	20	ralbidv 3161	. . . 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 3411	. 2 ⊢ {𝑣 ∈ 𝐴 ∣ ∀𝑞 ∈ ran 𝑑 𝑞𝑅𝑣} = {𝑟 ∈ 𝐴 ∣ ∀𝑠 ∈ ran 𝑑 𝑠𝑅𝑟}
24		eqid 2737	. 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 2737	. 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 10424	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 848   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3401  Vcvv 3442   ⊆ wss 3903   class class class wbr 5100   ↦ cmpt 5181   Po wpo 5538   Or wor 5539  dom cdm 5632  ran crn 5633   “ cima 5635  ℩crio 7324  recscrecs 8312  cardccrd 9859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-en 8896  df-card 9863

This theorem is referenced by:  zorng  10426  zorn2  10428