Theorem zornlem5 3607

Description: Lemma for Zorn's lemma.

Hypotheses

Ref	Expression
zornlem.1	⊢ A ∈ V
zornlem.2	⊢ B = {f∣∃h ∈ On (f Fn h ∧ ∀t ∈ h (f ‘t) = (G ‘(f ↾ t)))}
zornlem.3	⊢ F = ∪B
zornlem.4	⊢ C = {z ∈ A∣∀g ∈ ran fgRz}
zornlem.5	⊢ D = {z ∈ A∣∀g ∈ (F “ x)gRz}
zornlem.6	⊢ G = {⟨f, t⟩∣t = ∪{v ∈ C∣∀u ∈ C ¬ uwv}}
zornlem.7	⊢ H = {z ∈ A∣∀g ∈ (F “ y)gRz}

Assertion

Ref	Expression
zornlem5	⊢ (((w We A ∧ x ∈ On) ∧ ∀y ∈ x ¬ H = ∅) → (F “ x) ⊆ A)

Distinct variable group(s):   x,y,w,h,t,z,f,g,u,v,A   B,h,t,f   x,F,y,z,v,u,f,g,h,t   h,G,t,f   t,C   y,D,u,v,f,t   x,R,y,z,w,g,u,v,f,t   x,H,u,v,f,t

Proof of Theorem zornlem5

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 ⊢ ((w We A ∧ x ∈ On) → ∀y(w We A ∧ x ∈ On))
2		hbra1 1237	. . . . 5 ⊢ (∀y ∈ x ¬ H = ∅ → ∀y∀y ∈ x ¬ H = ∅)
3	1, 2	hban 704	. . . 4 ⊢ (((w We A ∧ x ∈ On) ∧ ∀y ∈ x ¬ H = ∅) → ∀y((w We A ∧ x ∈ On) ∧ ∀y ∈ x ¬ H = ∅))
4		ax-17 925	. . . 4 ⊢ (s ∈ A → ∀y s ∈ A)
5		eleq1 1149	. . . . . . . . . . . . . 14 ⊢ ((F ‘y) = s → ((F ‘y) ∈ A ↔ s ∈ A))
6		zornlem.1	. . . . . . . . . . . . . . . 16 ⊢ A ∈ V
7		zornlem.2	. . . . . . . . . . . . . . . 16 ⊢ B = {f∣∃h ∈ On (f Fn h ∧ ∀t ∈ h (f ‘t) = (G ‘(f ↾ t)))}
8		zornlem.3	. . . . . . . . . . . . . . . 16 ⊢ F = ∪B
9		zornlem.4	. . . . . . . . . . . . . . . 16 ⊢ C = {z ∈ A∣∀g ∈ ran fgRz}
10		zornlem.7	. . . . . . . . . . . . . . . 16 ⊢ H = {z ∈ A∣∀g ∈ (F “ y)gRz}
11		zornlem.6	. . . . . . . . . . . . . . . 16 ⊢ G = {⟨f, t⟩∣t = ∪{v ∈ C∣∀u ∈ C ¬ uwv}}
12	6, 7, 8, 9, 10, 11	zornlem1 3603	. . . . . . . . . . . . . . 15 ⊢ ((y ∈ On ∧ (w We A ∧ ¬ H = ∅)) → (F ‘y) ∈ H)
13		ssrab 1556	. . . . . . . . . . . . . . . . 17 ⊢ {z ∈ A∣∀g ∈ (F “ y)gRz} ⊆ A
14	10, 13	eqsstr 1530	. . . . . . . . . . . . . . . 16 ⊢ H ⊆ A
15	14	sseli 1504	. . . . . . . . . . . . . . 15 ⊢ ((F ‘y) ∈ H → (F ‘y) ∈ A)
16	12, 15	syl 12	. . . . . . . . . . . . . 14 ⊢ ((y ∈ On ∧ (w We A ∧ ¬ H = ∅)) → (F ‘y) ∈ A)
17	5, 16	syl5bi 183	. . . . . . . . . . . . 13 ⊢ ((F ‘y) = s → ((y ∈ On ∧ (w We A ∧ ¬ H = ∅)) → s ∈ A))
18		onelon 2223	. . . . . . . . . . . . 13 ⊢ ((x ∈ On ∧ y ∈ x) → y ∈ On)
19	17, 18	sylani 356	. . . . . . . . . . . 12 ⊢ ((F ‘y) = s → (((x ∈ On ∧ y ∈ x) ∧ (w We A ∧ ¬ H = ∅)) → s ∈ A))
20	19	com12 13	. . . . . . . . . . 11 ⊢ (((x ∈ On ∧ y ∈ x) ∧ (w We A ∧ ¬ H = ∅)) → ((F ‘y) = s → s ∈ A))
21	20	exp43 301	. . . . . . . . . 10 ⊢ (x ∈ On → (y ∈ x → (w We A → (¬ H = ∅ → ((F ‘y) = s → s ∈ A)))))
22	21	com3r 35	. . . . . . . . 9 ⊢ (w We A → (x ∈ On → (y ∈ x → (¬ H = ∅ → ((F ‘y) = s → s ∈ A)))))
23	22	imp 277	. . . . . . . 8 ⊢ ((w We A ∧ x ∈ On) → (y ∈ x → (¬ H = ∅ → ((F ‘y) = s → s ∈ A))))
24	23	a2d 15	. . . . . . 7 ⊢ ((w We A ∧ x ∈ On) → ((y ∈ x → ¬ H = ∅) → (y ∈ x → ((F ‘y) = s → s ∈ A))))
25	24	a4sd 683	. . . . . 6 ⊢ ((w We A ∧ x ∈ On) → (∀y(y ∈ x → ¬ H = ∅) → (y ∈ x → ((F ‘y) = s → s ∈ A))))
26		df-ral 1205	. . . . . 6 ⊢ (∀y ∈ x ¬ H = ∅ ↔ ∀y(y ∈ x → ¬ H = ∅))
27	25, 26	syl5ib 181	. . . . 5 ⊢ ((w We A ∧ x ∈ On) → (∀y ∈ x ¬ H = ∅ → (y ∈ x → ((F ‘y) = s → s ∈ A))))
28	27	imp 277	. . . 4 ⊢ (((w We A ∧ x ∈ On) ∧ ∀y ∈ x ¬ H = ∅) → (y ∈ x → ((F ‘y) = s → s ∈ A)))
29	3, 4, 28	r19.23ad 1285	. . 3 ⊢ (((w We A ∧ x ∈ On) ∧ ∀y ∈ x ¬ H = ∅) → (∃y ∈ x (F ‘y) = s → s ∈ A))
30	7, 8	tfrlem7 2955	. . . 4 ⊢ Fun F
31		fvelima 2859	. . . 4 ⊢ ((Fun F ∧ s ∈ (F “ x)) → ∃y ∈ x (F ‘y) = s)
32	30, 31	mpan 518	. . 3 ⊢ (s ∈ (F “ x) → ∃y ∈ x (F ‘y) = s)
33	29, 32	syl5 22	. 2 ⊢ (((w We A ∧ x ∈ On) ∧ ∀y ∈ x ¬ H = ∅) → (s ∈ (F “ x) → s ∈ A))
34	33	ssrdv 1509	1 ⊢ (((w We A ∧ x ∈ On) ∧ ∀y ∈ x ¬ H = ∅) → (F “ x) ⊆ A)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  {crab 1204  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  ∪cuni 1919   class class class wbr 2054  {copab 2055   We wwe 2062  Oncon0 2199  ran crn 2411   ↾ cres 2412   “ cima 2413  Fun wfun 2416   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  zornlem6 3608  zornlem7 3609

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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