Theorem zorn2lem4 10504

Description: Lemma for zorn2 10511. (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		pm3.24 408	. 2 ⊢ ¬ (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)
2		df-ne 2958	. . . . 5 ⊢ (𝐷 ≠ ∅ ↔ ¬ 𝐷 = ∅)
3	2	ralbii 3110	. . . 4 ⊢ (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥 ∈ On ¬ 𝐷 = ∅)
4		df-ral 3079	. . . 4 ⊢ (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
5		ralnex 3090	. . . 4 ⊢ (∀𝑥 ∈ On ¬ 𝐷 = ∅ ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
6	3, 4, 5	3bitr3i 304	. . 3 ⊢ (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
7		weso 5650	. . . . . . . . 9 ⊢ (𝑤 We 𝐴 → 𝑤 Or 𝐴)
8	7	adantr 486	. . . . . . . 8 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → 𝑤 Or 𝐴)
9		vex 3457	. . . . . . . 8 ⊢ 𝑤 ∈ V
10		soex 7921	. . . . . . . 8 ⊢ ((𝑤 Or 𝐴 ∧ 𝑤 ∈ V) → 𝐴 ∈ V)
11	8, 9, 10	sylancl 598	. . . . . . 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 8389	. . . . . . . . . 10 ⊢ 𝐹 Fn On
14		fvelrnb 6942	. . . . . . . . . 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 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑤 We 𝐴
17		nfa1 2188	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)
18	16, 17	nfan 1932	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
19		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
20		zorn2lem.5	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}
21	20	ssrab3 4033	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 ⊆ 𝐴
22		zorn2lem.4	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
23	12, 22, 20	zorn2lem1 10501	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐷)
24	21, 23	sselid 3932	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐴)
25		eleq1 2850	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
26	24, 25	syl5ibcom 248	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))
27	26	exp32 426	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ On → (𝑤 We 𝐴 → (𝐷 ≠ ∅ → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))))
28	27	com12 33	. . . . . . . . . . . . 13 ⊢ (𝑤 We 𝐴 → (𝑥 ∈ On → (𝐷 ≠ ∅ → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))))
29	28	a2d 30	. . . . . . . . . . . 12 ⊢ (𝑤 We 𝐴 → ((𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))))
30	29	spsd 2225	. . . . . . . . . . 11 ⊢ (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))))
31	30	imp 412	. . . . . . . . . 10 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴)))
32	18, 19, 31	rexlimd 3271	. . . . . . . . 9 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (∃𝑥 ∈ On (𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))
33	15, 32	biimtrid 245	. . . . . . . 8 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ 𝐴))
34	33	ssrdv 3940	. . . . . . 7 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹 ⊆ 𝐴)
35	11, 34	ssexd 5293	. . . . . 6 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹 ∈ V)
36	35	ex 418	. . . . 5 ⊢ (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
37	36	adantl 487	. . . 4 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
38	12, 22, 20	zorn2lem3 10503	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅))) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
39	38	exp45 444	. . . . . . . . . . . . 13 ⊢ (𝑅 Po 𝐴 → (𝑥 ∈ On → (𝑤 We 𝐴 → (𝐷 ≠ ∅ → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))))
40	39	com23 87	. . . . . . . . . . . 12 ⊢ (𝑅 Po 𝐴 → (𝑤 We 𝐴 → (𝑥 ∈ On → (𝐷 ≠ ∅ → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))))
41	40	imp 412	. . . . . . . . . . 11 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (𝑥 ∈ On → (𝐷 ≠ ∅ → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))))
42	41	a2d 30	. . . . . . . . . 10 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))))
43	42	imp4a 428	. . . . . . . . 9 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
44	43	alrimdv 1962	. . . . . . . 8 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
45	44	alimdv 1949	. . . . . . 7 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥∀𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
46		r2al 3200	. . . . . . 7 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) ↔ ∀𝑥∀𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
47	45, 46	imbitrrdi 255	. . . . . 6 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
48		ssid 3956	. . . . . . . 8 ⊢ On ⊆ On
49	13	tz7.48lem 8433	. . . . . . . 8 ⊢ ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ◡(𝐹 ↾ On))
50	48, 49	mpan 703	. . . . . . 7 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → Fun ◡(𝐹 ↾ On))
51		fnrel 6638	. . . . . . . . . . 11 ⊢ (𝐹 Fn On → Rel 𝐹)
52	13, 51	ax-mp 5	. . . . . . . . . 10 ⊢ Rel 𝐹
53	13	fndmi 6640	. . . . . . . . . . 11 ⊢ dom 𝐹 = On
54	53	eqimssi 3994	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ On
55		relssres 6019	. . . . . . . . . 10 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
56	52, 54, 55	mp2an 705	. . . . . . . . 9 ⊢ (𝐹 ↾ On) = 𝐹
57	56	cnveqi 5858	. . . . . . . 8 ⊢ ◡(𝐹 ↾ On) = ◡𝐹
58	57	funeqi 6558	. . . . . . 7 ⊢ (Fun ◡(𝐹 ↾ On) ↔ Fun ◡𝐹)
59	50, 58	sylib 221	. . . . . 6 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → Fun ◡𝐹)
60	47, 59	syl6 36	. . . . 5 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → Fun ◡𝐹))
61		onprc 7780	. . . . . 6 ⊢ ¬ On ∈ V
62		funrnex 7954	. . . . . . . 8 ⊢ (dom ◡𝐹 ∈ V → (Fun ◡𝐹 → ran ◡𝐹 ∈ V))
63	62	com12 33	. . . . . . 7 ⊢ (Fun ◡𝐹 → (dom ◡𝐹 ∈ V → ran ◡𝐹 ∈ V))
64		df-rn 5670	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
65	64	eleq1i 2853	. . . . . . 7 ⊢ (ran 𝐹 ∈ V ↔ dom ◡𝐹 ∈ V)
66		dfdm4 5883	. . . . . . . . 9 ⊢ dom 𝐹 = ran ◡𝐹
67	53, 66	eqtr3i 2787	. . . . . . . 8 ⊢ On = ran ◡𝐹
68	67	eleq1i 2853	. . . . . . 7 ⊢ (On ∈ V ↔ ran ◡𝐹 ∈ V)
69	63, 65, 68	3imtr4g 299	. . . . . 6 ⊢ (Fun ◡𝐹 → (ran 𝐹 ∈ V → On ∈ V))
70	61, 69	mtoi 202	. . . . 5 ⊢ (Fun ◡𝐹 → ¬ ran 𝐹 ∈ V)
71	60, 70	syl6 36	. . . 4 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ¬ ran 𝐹 ∈ V))
72	37, 71	jcad 522	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)))
73	6, 72	biimtrrid 246	. 2 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (¬ ∃𝑥 ∈ On 𝐷 = ∅ → (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)))
74	1, 73	mt3i 150	1 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ∃𝑥 ∈ On 𝐷 = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3414  Vcvv 3453   ⊆ wss 3902  ∅c0 4282   class class class wbr 5107   ↦ cmpt 5190   Po wpo 5565   Or wor 5566   We wwe 5611  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   ↾ cres 5661   “ cima 5662  Rel wrel 5664  Oncon0 6361  Fun wfun 6531   Fn wfn 6532  ‘cfv 6537  ℩crio 7372  recscrecs 8362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  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 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363

This theorem is used by:  zorn2lem7  10507