Theorem zorn2lem4 10568

Description: Lemma for zorn2 10575. (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 2947	. . . . 5 ⊢ (𝐷 ≠ ∅ ↔ ¬ 𝐷 = ∅)
3	2	ralbii 3099	. . . 4 ⊢ (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥 ∈ On ¬ 𝐷 = ∅)
4		df-ral 3068	. . . 4 ⊢ (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
5		ralnex 3078	. . . 4 ⊢ (∀𝑥 ∈ On ¬ 𝐷 = ∅ ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
6	3, 4, 5	3bitr3i 301	. . 3 ⊢ (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
7		weso 5691	. . . . . . . . 9 ⊢ (𝑤 We 𝐴 → 𝑤 Or 𝐴)
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → 𝑤 Or 𝐴)
9		vex 3492	. . . . . . . 8 ⊢ 𝑤 ∈ V
10		soex 7961	. . . . . . . 8 ⊢ ((𝑤 Or 𝐴 ∧ 𝑤 ∈ V) → 𝐴 ∈ V)
11	8, 9, 10	sylancl 585	. . . . . . 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 8453	. . . . . . . . . 10 ⊢ 𝐹 Fn On
14		fvelrnb 6982	. . . . . . . . . 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 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑤 We 𝐴
17		nfa1 2152	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)
18	16, 17	nfan 1898	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
19		nfv 1913	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
20		zorn2lem.5	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}
21	20	ssrab3 4105	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 ⊆ 𝐴
22		zorn2lem.4	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
23	12, 22, 20	zorn2lem1 10565	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐷)
24	21, 23	sselid 4006	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐴)
25		eleq1 2832	. . . . . . . . . . . . . . . 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 3272	. . . . . . . . 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 4014	. . . . . . 7 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹 ⊆ 𝐴)
35	11, 34	ssexd 5342	. . . . . 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 10567	. . . . . . . . . . . . . 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 1928	. . . . . . . 8 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
45	44	alimdv 1915	. . . . . . 7 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥∀𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
46		r2al 3201	. . . . . . 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 4031	. . . . . . . 8 ⊢ On ⊆ On
49	13	tz7.48lem 8497	. . . . . . . 8 ⊢ ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ◡(𝐹 ↾ On))
50	48, 49	mpan 689	. . . . . . 7 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → Fun ◡(𝐹 ↾ On))
51		fnrel 6681	. . . . . . . . . . 11 ⊢ (𝐹 Fn On → Rel 𝐹)
52	13, 51	ax-mp 5	. . . . . . . . . 10 ⊢ Rel 𝐹
53	13	fndmi 6683	. . . . . . . . . . 11 ⊢ dom 𝐹 = On
54	53	eqimssi 4069	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ On
55		relssres 6051	. . . . . . . . . 10 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
56	52, 54, 55	mp2an 691	. . . . . . . . 9 ⊢ (𝐹 ↾ On) = 𝐹
57	56	cnveqi 5899	. . . . . . . 8 ⊢ ◡(𝐹 ↾ On) = ◡𝐹
58	57	funeqi 6599	. . . . . . 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 7813	. . . . . 6 ⊢ ¬ On ∈ V
62		funrnex 7994	. . . . . . . 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 5711	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
65	64	eleq1i 2835	. . . . . . 7 ⊢ (ran 𝐹 ∈ V ↔ dom ◡𝐹 ∈ V)
66		dfdm4 5920	. . . . . . . . 9 ⊢ dom 𝐹 = ran ◡𝐹
67	53, 66	eqtr3i 2770	. . . . . . . 8 ⊢ On = ran ◡𝐹
68	67	eleq1i 2835	. . . . . . 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 1535   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  {crab 3443  Vcvv 3488   ⊆ wss 3976  ∅c0 4352   class class class wbr 5166   ↦ cmpt 5249   Po wpo 5605   Or wor 5606   We wwe 5651  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703  Rel wrel 5705  Oncon0 6395  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573  ℩crio 7403  recscrecs 8426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427

This theorem is referenced by:  zorn2lem7  10571