Theorem zorn2lem4 10518

Description: Lemma for zorn2 10525. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)

Hypotheses

Ref	Expression
zorn2lem.3	⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)))
zorn2lem.4	⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
zorn2lem.5	⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}

Assertion

Ref	Expression
zorn2lem4	⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ∃𝑥 ∈ On 𝐷 = ∅)

Distinct variable groups:   𝑓,𝑔,𝑢,𝑣,𝑤,𝑥,𝑧,𝐴   𝐷,𝑓,𝑢,𝑣   𝑓,𝐹,𝑔,𝑢,𝑣,𝑥,𝑧   𝑅,𝑓,𝑔,𝑢,𝑣,𝑤,𝑥,𝑧   𝑣,𝐶

Allowed substitution hints:   𝐶(𝑥,𝑧,𝑤,𝑢,𝑓,𝑔)   𝐷(𝑥,𝑧,𝑤,𝑔)   𝐹(𝑤)

Proof of Theorem zorn2lem4

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm3.24 402	. 2 ⊢ ¬ (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)
2		df-ne 2934	. . . . 5 ⊢ (𝐷 ≠ ∅ ↔ ¬ 𝐷 = ∅)
3	2	ralbii 3083	. . . 4 ⊢ (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥 ∈ On ¬ 𝐷 = ∅)
4		df-ral 3053	. . . 4 ⊢ (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
5		ralnex 3063	. . . 4 ⊢ (∀𝑥 ∈ On ¬ 𝐷 = ∅ ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
6	3, 4, 5	3bitr3i 301	. . 3 ⊢ (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
7		weso 5650	. . . . . . . . 9 ⊢ (𝑤 We 𝐴 → 𝑤 Or 𝐴)
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → 𝑤 Or 𝐴)
9		vex 3468	. . . . . . . 8 ⊢ 𝑤 ∈ V
10		soex 7922	. . . . . . . 8 ⊢ ((𝑤 Or 𝐴 ∧ 𝑤 ∈ V) → 𝐴 ∈ V)
11	8, 9, 10	sylancl 586	. . . . . . 7 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → 𝐴 ∈ V)
12		zorn2lem.3	. . . . . . . . . . 11 ⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣)))
13	12	tfr1 8416	. . . . . . . . . 10 ⊢ 𝐹 Fn On
14		fvelrnb 6944	. . . . . . . . . 10 ⊢ (𝐹 Fn On → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ On (𝐹‘𝑥) = 𝑦))
15	13, 14	ax-mp 5	. . . . . . . . 9 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ On (𝐹‘𝑥) = 𝑦)
16		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑤 We 𝐴
17		nfa1 2152	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)
18	16, 17	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
19		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
20		zorn2lem.5	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}
21	20	ssrab3 4062	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 ⊆ 𝐴
22		zorn2lem.4	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
23	12, 22, 20	zorn2lem1 10515	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐷)
24	21, 23	sselid 3961	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐴)
25		eleq1 2823	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
26	24, 25	syl5ibcom 245	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))
27	26	exp32 420	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ On → (𝑤 We 𝐴 → (𝐷 ≠ ∅ → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))))
28	27	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑤 We 𝐴 → (𝑥 ∈ On → (𝐷 ≠ ∅ → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))))
29	28	a2d 29	. . . . . . . . . . . 12 ⊢ (𝑤 We 𝐴 → ((𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))))
30	29	spsd 2188	. . . . . . . . . . 11 ⊢ (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))))
31	30	imp 406	. . . . . . . . . 10 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴)))
32	18, 19, 31	rexlimd 3253	. . . . . . . . 9 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (∃𝑥 ∈ On (𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))
33	15, 32	biimtrid 242	. . . . . . . 8 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ 𝐴))
34	33	ssrdv 3969	. . . . . . 7 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹 ⊆ 𝐴)
35	11, 34	ssexd 5299	. . . . . 6 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹 ∈ V)
36	35	ex 412	. . . . 5 ⊢ (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
37	36	adantl 481	. . . 4 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
38	12, 22, 20	zorn2lem3 10517	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅))) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
39	38	exp45 438	. . . . . . . . . . . . 13 ⊢ (𝑅 Po 𝐴 → (𝑥 ∈ On → (𝑤 We 𝐴 → (𝐷 ≠ ∅ → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))))
40	39	com23 86	. . . . . . . . . . . 12 ⊢ (𝑅 Po 𝐴 → (𝑤 We 𝐴 → (𝑥 ∈ On → (𝐷 ≠ ∅ → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))))
41	40	imp 406	. . . . . . . . . . 11 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (𝑥 ∈ On → (𝐷 ≠ ∅ → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))))
42	41	a2d 29	. . . . . . . . . 10 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))))
43	42	imp4a 422	. . . . . . . . 9 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
44	43	alrimdv 1929	. . . . . . . 8 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
45	44	alimdv 1916	. . . . . . 7 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥∀𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
46		r2al 3181	. . . . . . 7 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) ↔ ∀𝑥∀𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
47	45, 46	imbitrrdi 252	. . . . . 6 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
48		ssid 3986	. . . . . . . 8 ⊢ On ⊆ On
49	13	tz7.48lem 8460	. . . . . . . 8 ⊢ ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ◡(𝐹 ↾ On))
50	48, 49	mpan 690	. . . . . . 7 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → Fun ◡(𝐹 ↾ On))
51		fnrel 6645	. . . . . . . . . . 11 ⊢ (𝐹 Fn On → Rel 𝐹)
52	13, 51	ax-mp 5	. . . . . . . . . 10 ⊢ Rel 𝐹
53	13	fndmi 6647	. . . . . . . . . . 11 ⊢ dom 𝐹 = On
54	53	eqimssi 4024	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ On
55		relssres 6014	. . . . . . . . . 10 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
56	52, 54, 55	mp2an 692	. . . . . . . . 9 ⊢ (𝐹 ↾ On) = 𝐹
57	56	cnveqi 5859	. . . . . . . 8 ⊢ ◡(𝐹 ↾ On) = ◡𝐹
58	57	funeqi 6562	. . . . . . 7 ⊢ (Fun ◡(𝐹 ↾ On) ↔ Fun ◡𝐹)
59	50, 58	sylib 218	. . . . . 6 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → Fun ◡𝐹)
60	47, 59	syl6 35	. . . . 5 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → Fun ◡𝐹))
61		onprc 7777	. . . . . 6 ⊢ ¬ On ∈ V
62		funrnex 7957	. . . . . . . 8 ⊢ (dom ◡𝐹 ∈ V → (Fun ◡𝐹 → ran ◡𝐹 ∈ V))
63	62	com12 32	. . . . . . 7 ⊢ (Fun ◡𝐹 → (dom ◡𝐹 ∈ V → ran ◡𝐹 ∈ V))
64		df-rn 5670	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
65	64	eleq1i 2826	. . . . . . 7 ⊢ (ran 𝐹 ∈ V ↔ dom ◡𝐹 ∈ V)
66		dfdm4 5880	. . . . . . . . 9 ⊢ dom 𝐹 = ran ◡𝐹
67	53, 66	eqtr3i 2761	. . . . . . . 8 ⊢ On = ran ◡𝐹
68	67	eleq1i 2826	. . . . . . 7 ⊢ (On ∈ V ↔ ran ◡𝐹 ∈ V)
69	63, 65, 68	3imtr4g 296	. . . . . 6 ⊢ (Fun ◡𝐹 → (ran 𝐹 ∈ V → On ∈ V))
70	61, 69	mtoi 199	. . . . 5 ⊢ (Fun ◡𝐹 → ¬ ran 𝐹 ∈ V)
71	60, 70	syl6 35	. . . 4 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ¬ ran 𝐹 ∈ V))
72	37, 71	jcad 512	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)))
73	6, 72	biimtrrid 243	. 2 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (¬ ∃𝑥 ∈ On 𝐷 = ∅ → (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)))
74	1, 73	mt3i 149	1 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ∃𝑥 ∈ On 𝐷 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  {crab 3420  Vcvv 3464   ⊆ wss 3931  ∅c0 4313   class class class wbr 5124   ↦ cmpt 5206   Po wpo 5564   Or wor 5565   We wwe 5610  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   ↾ cres 5661   “ cima 5662  Rel wrel 5664  Oncon0 6357  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  ℩crio 7366  recscrecs 8389

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-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-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-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-riota 7367  df-ov 7413  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390

This theorem is referenced by:  zorn2lem7  10521