Theorem zorn2lem4 10435

Description: Lemma for zorn2 10442. (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		pm3.24 403	. 2 ⊢ ¬ (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)
2		df-ne 2944	. . . . 5 ⊢ (𝐷 ≠ ∅ ↔ ¬ 𝐷 = ∅)
3	2	ralbii 3096	. . . 4 ⊢ (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥 ∈ On ¬ 𝐷 = ∅)
4		df-ral 3065	. . . 4 ⊢ (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
5		ralnex 3075	. . . 4 ⊢ (∀𝑥 ∈ On ¬ 𝐷 = ∅ ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
6	3, 4, 5	3bitr3i 300	. . 3 ⊢ (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
7		weso 5624	. . . . . . . . 9 ⊢ (𝑤 We 𝐴 → 𝑤 Or 𝐴)
8	7	adantr 481	. . . . . . . 8 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → 𝑤 Or 𝐴)
9		vex 3449	. . . . . . . 8 ⊢ 𝑤 ∈ V
10		soex 7858	. . . . . . . 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 8343	. . . . . . . . . 10 ⊢ 𝐹 Fn On
14		fvelrnb 6903	. . . . . . . . . 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 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑤 We 𝐴
17		nfa1 2148	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)
18	16, 17	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
19		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
20		zorn2lem.5	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}
21	20	ssrab3 4040	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 ⊆ 𝐴
22		zorn2lem.4	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
23	12, 22, 20	zorn2lem1 10432	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐷)
24	21, 23	sselid 3942	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐴)
25		eleq1 2825	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
26	24, 25	syl5ibcom 244	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))
27	26	exp32 421	. . . . . . . . . . . . . 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 2180	. . . . . . . . . . 11 ⊢ (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))))
31	30	imp 407	. . . . . . . . . 10 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴)))
32	18, 19, 31	rexlimd 3249	. . . . . . . . 9 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (∃𝑥 ∈ On (𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))
33	15, 32	biimtrid 241	. . . . . . . 8 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ 𝐴))
34	33	ssrdv 3950	. . . . . . 7 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹 ⊆ 𝐴)
35	11, 34	ssexd 5281	. . . . . 6 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → ran 𝐹 ∈ V)
36	35	ex 413	. . . . 5 ⊢ (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
37	36	adantl 482	. . . 4 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
38	12, 22, 20	zorn2lem3 10434	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅))) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
39	38	exp45 439	. . . . . . . . . . . . 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 407	. . . . . . . . . . 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 423	. . . . . . . . 9 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
44	43	alrimdv 1932	. . . . . . . 8 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
45	44	alimdv 1919	. . . . . . 7 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥∀𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
46		r2al 3191	. . . . . . 7 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) ↔ ∀𝑥∀𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
47	45, 46	syl6ibr 251	. . . . . 6 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
48		ssid 3966	. . . . . . . 8 ⊢ On ⊆ On
49	13	tz7.48lem 8387	. . . . . . . 8 ⊢ ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ◡(𝐹 ↾ On))
50	48, 49	mpan 688	. . . . . . 7 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → Fun ◡(𝐹 ↾ On))
51		fnrel 6604	. . . . . . . . . . 11 ⊢ (𝐹 Fn On → Rel 𝐹)
52	13, 51	ax-mp 5	. . . . . . . . . 10 ⊢ Rel 𝐹
53	13	fndmi 6606	. . . . . . . . . . 11 ⊢ dom 𝐹 = On
54	53	eqimssi 4002	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ On
55		relssres 5978	. . . . . . . . . 10 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
56	52, 54, 55	mp2an 690	. . . . . . . . 9 ⊢ (𝐹 ↾ On) = 𝐹
57	56	cnveqi 5830	. . . . . . . 8 ⊢ ◡(𝐹 ↾ On) = ◡𝐹
58	57	funeqi 6522	. . . . . . 7 ⊢ (Fun ◡(𝐹 ↾ On) ↔ Fun ◡𝐹)
59	50, 58	sylib 217	. . . . . 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 7712	. . . . . 6 ⊢ ¬ On ∈ V
62		funrnex 7886	. . . . . . . 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 5644	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
65	64	eleq1i 2828	. . . . . . 7 ⊢ (ran 𝐹 ∈ V ↔ dom ◡𝐹 ∈ V)
66		dfdm4 5851	. . . . . . . . 9 ⊢ dom 𝐹 = ran ◡𝐹
67	53, 66	eqtr3i 2766	. . . . . . . 8 ⊢ On = ran ◡𝐹
68	67	eleq1i 2828	. . . . . . 7 ⊢ (On ∈ V ↔ ran ◡𝐹 ∈ V)
69	63, 65, 68	3imtr4g 295	. . . . . 6 ⊢ (Fun ◡𝐹 → (ran 𝐹 ∈ V → On ∈ V))
70	61, 69	mtoi 198	. . . . 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 513	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)))
73	6, 72	biimtrrid 242	. 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 205   ∧ wa 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  {crab 3407  Vcvv 3445   ⊆ wss 3910  ∅c0 4282   class class class wbr 5105   ↦ cmpt 5188   Po wpo 5543   Or wor 5544   We wwe 5587  ^◡ccnv 5632  dom cdm 5633  ran crn 5634   ↾ cres 5635   “ cima 5636  Rel wrel 5638  Oncon0 6317  Fun wfun 6490   Fn wfn 6491  ‘cfv 6496  ℩crio 7312  recscrecs 8316

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317

This theorem is referenced by:  zorn2lem7  10438