Theorem zorn2lem4 10457

Description: Lemma for zorn2 10464. (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		pm3.24 406	. 2 ⊢ ¬ (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)
2		df-ne 2959	. . . . 5 ⊢ (𝐷 ≠ ∅ ↔ ¬ 𝐷 = ∅)
3	2	ralbii 3109	. . . 4 ⊢ (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥 ∈ On ¬ 𝐷 = ∅)
4		df-ral 3078	. . . 4 ⊢ (∀𝑥 ∈ On 𝐷 ≠ ∅ ↔ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
5		ralnex 3089	. . . 4 ⊢ (∀𝑥 ∈ On ¬ 𝐷 = ∅ ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
6	3, 4, 5	3bitr3i 303	. . 3 ⊢ (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) ↔ ¬ ∃𝑥 ∈ On 𝐷 = ∅)
7		weso 5639	. . . . . . . . 9 ⊢ (𝑤 We 𝐴 → 𝑤 Or 𝐴)
8	7	adantr 484	. . . . . . . 8 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → 𝑤 Or 𝐴)
9		vex 3459	. . . . . . . 8 ⊢ 𝑤 ∈ V
10		soex 7903	. . . . . . . 8 ⊢ ((𝑤 Or 𝐴 ∧ 𝑤 ∈ V) → 𝐴 ∈ V)
11	8, 9, 10	sylancl 595	. . . . . . 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 8369	. . . . . . . . . 10 ⊢ 𝐹 Fn On
14		fvelrnb 6928	. . . . . . . . . 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 1935	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑤 We 𝐴
17		nfa1 2186	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)
18	16, 17	nfan 1920	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅))
19		nfv 1935	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
20		zorn2lem.5	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧}
21	20	ssrab3 4036	. . . . . . . . . . . . . . . . 17 ⊢ 𝐷 ⊆ 𝐴
22		zorn2lem.4	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}
23	12, 22, 20	zorn2lem1 10454	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐷)
24	21, 23	sselid 3935	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐴)
25		eleq1 2851	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
26	24, 25	syl5ibcom 247	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))
27	26	exp32 424	. . . . . . . . . . . . . 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 2223	. . . . . . . . . . 11 ⊢ (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))))
31	30	imp 410	. . . . . . . . . 10 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑥 ∈ On → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴)))
32	18, 19, 31	rexlimd 3270	. . . . . . . . 9 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (∃𝑥 ∈ On (𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝐴))
33	15, 32	biimtrid 244	. . . . . . . 8 ⊢ ((𝑤 We 𝐴 ∧ ∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅)) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ 𝐴))
34	33	ssrdv 3943	. . . . . . 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 416	. . . . 5 ⊢ (𝑤 We 𝐴 → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
37	36	adantl 485	. . . 4 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ran 𝐹 ∈ V))
38	12, 22, 20	zorn2lem3 10456	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅))) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
39	38	exp45 442	. . . . . . . . . . . . 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 410	. . . . . . . . . . 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 426	. . . . . . . . 9 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
44	43	alrimdv 1950	. . . . . . . 8 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ((𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
45	44	alimdv 1937	. . . . . . 7 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥∀𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))))
46		r2al 3199	. . . . . . 7 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) ↔ ∀𝑥∀𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
47	45, 46	imbitrrdi 254	. . . . . 6 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)))
48		ssid 3959	. . . . . . . 8 ⊢ On ⊆ On
49	13	tz7.48lem 8413	. . . . . . . 8 ⊢ ((On ⊆ On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦)) → Fun ◡(𝐹 ↾ On))
50	48, 49	mpan 700	. . . . . . 7 ⊢ (∀𝑥 ∈ On ∀𝑦 ∈ 𝑥 ¬ (𝐹‘𝑥) = (𝐹‘𝑦) → Fun ◡(𝐹 ↾ On))
51		fnrel 6624	. . . . . . . . . . 11 ⊢ (𝐹 Fn On → Rel 𝐹)
52	13, 51	ax-mp 5	. . . . . . . . . 10 ⊢ Rel 𝐹
53	13	fndmi 6626	. . . . . . . . . . 11 ⊢ dom 𝐹 = On
54	53	eqimssi 3997	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ On
55		relssres 6009	. . . . . . . . . 10 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
56	52, 54, 55	mp2an 702	. . . . . . . . 9 ⊢ (𝐹 ↾ On) = 𝐹
57	56	cnveqi 5847	. . . . . . . 8 ⊢ ◡(𝐹 ↾ On) = ◡𝐹
58	57	funeqi 6543	. . . . . . 7 ⊢ (Fun ◡(𝐹 ↾ On) ↔ Fun ◡𝐹)
59	50, 58	sylib 220	. . . . . 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 7762	. . . . . 6 ⊢ ¬ On ∈ V
62		funrnex 7936	. . . . . . . 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 5659	. . . . . . . 8 ⊢ ran 𝐹 = dom ◡𝐹
65	64	eleq1i 2854	. . . . . . 7 ⊢ (ran 𝐹 ∈ V ↔ dom ◡𝐹 ∈ V)
66		dfdm4 5872	. . . . . . . . 9 ⊢ dom 𝐹 = ran ◡𝐹
67	53, 66	eqtr3i 2788	. . . . . . . 8 ⊢ On = ran ◡𝐹
68	67	eleq1i 2854	. . . . . . 7 ⊢ (On ∈ V ↔ ran ◡𝐹 ∈ V)
69	63, 65, 68	3imtr4g 298	. . . . . 6 ⊢ (Fun ◡𝐹 → (ran 𝐹 ∈ V → On ∈ V))
70	61, 69	mtoi 201	. . . . 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 520	. . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → (∀𝑥(𝑥 ∈ On → 𝐷 ≠ ∅) → (ran 𝐹 ∈ V ∧ ¬ ran 𝐹 ∈ V)))
73	6, 72	biimtrrid 245	. 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 208   ∧ wa 399  ∀wal 1559   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ∀wral 3077  ∃wrex 3087  {crab 3415  Vcvv 3455   ⊆ wss 3905  ∅c0 4286   class class class wbr 5101   ↦ cmpt 5182   Po wpo 5554   Or wor 5555   We wwe 5600  ^◡ccnv 5647  dom cdm 5648  ran crn 5649   ↾ cres 5650   “ cima 5651  Rel wrel 5653  Oncon0 6347  Fun wfun 6516   Fn wfn 6517  ‘cfv 6522  ℩crio 7353  recscrecs 8342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343

This theorem is referenced by:  zorn2lem7  10460