Theorem onint 2261

Description: The intersection (infimum) of a non-empty class of ordinal numbers belongs to the class. Compare Exercise 4 of [TakeutiZaring] p. 45.

Assertion

Ref	Expression
onint	⊢ ((A ⊆ On ∧ ¬ A = ∅) → ∩A ∈ A)

Proof of Theorem onint

Step	Hyp	Ref	Expression
1		ssel 1502	. . . . . . . . . . . . . . . . . . . 20 ⊢ (A ⊆ On → (z ∈ A → z ∈ On))
2		ontri1 2232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((x ∈ On ∧ z ∈ On) → (x ⊆ z ↔ ¬ z ∈ x))
3		ssel 1502	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (x ⊆ z → (y ∈ x → y ∈ z))
4	2, 3	syl6bir 188	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((x ∈ On ∧ z ∈ On) → (¬ z ∈ x → (y ∈ x → y ∈ z)))
5	4	exp 291	. . . . . . . . . . . . . . . . . . . 20 ⊢ (x ∈ On → (z ∈ On → (¬ z ∈ x → (y ∈ x → y ∈ z))))
6	1, 5	sylan9 359	. . . . . . . . . . . . . . . . . . 19 ⊢ ((A ⊆ On ∧ x ∈ On) → (z ∈ A → (¬ z ∈ x → (y ∈ x → y ∈ z))))
7	6	com4r 41	. . . . . . . . . . . . . . . . . 18 ⊢ (y ∈ x → ((A ⊆ On ∧ x ∈ On) → (z ∈ A → (¬ z ∈ x → y ∈ z))))
8	7	imp31 280	. . . . . . . . . . . . . . . . 17 ⊢ (((y ∈ x ∧ (A ⊆ On ∧ x ∈ On)) ∧ z ∈ A) → (¬ z ∈ x → y ∈ z))
9	8	r19.20dva 1256	. . . . . . . . . . . . . . . 16 ⊢ ((y ∈ x ∧ (A ⊆ On ∧ x ∈ On)) → (∀z ∈ A ¬ z ∈ x → ∀z ∈ A y ∈ z))
10		disj 1733	. . . . . . . . . . . . . . . 16 ⊢ ((A ∩ x) = ∅ ↔ ∀z ∈ A ¬ z ∈ x)
11		visset 1350	. . . . . . . . . . . . . . . . 17 ⊢ y ∈ V
12	11	elint2 1972	. . . . . . . . . . . . . . . 16 ⊢ (y ∈ ∩A ↔ ∀z ∈ A y ∈ z)
13	9, 10, 12	3imtr4g 426	. . . . . . . . . . . . . . 15 ⊢ ((y ∈ x ∧ (A ⊆ On ∧ x ∈ On)) → ((A ∩ x) = ∅ → y ∈ ∩A))
14		ssel 1502	. . . . . . . . . . . . . . . 16 ⊢ (A ⊆ On → (x ∈ A → x ∈ On))
15	14	imdistani 340	. . . . . . . . . . . . . . 15 ⊢ ((A ⊆ On ∧ x ∈ A) → (A ⊆ On ∧ x ∈ On))
16	13, 15	sylan2 346	. . . . . . . . . . . . . 14 ⊢ ((y ∈ x ∧ (A ⊆ On ∧ x ∈ A)) → ((A ∩ x) = ∅ → y ∈ ∩A))
17	16	exp32 294	. . . . . . . . . . . . 13 ⊢ (y ∈ x → (A ⊆ On → (x ∈ A → ((A ∩ x) = ∅ → y ∈ ∩A))))
18	17	com4l 39	. . . . . . . . . . . 12 ⊢ (A ⊆ On → (x ∈ A → ((A ∩ x) = ∅ → (y ∈ x → y ∈ ∩A))))
19	18	imp32 281	. . . . . . . . . . 11 ⊢ ((A ⊆ On ∧ (x ∈ A ∧ (A ∩ x) = ∅)) → (y ∈ x → y ∈ ∩A))
20	19	ssrdv 1509	. . . . . . . . . 10 ⊢ ((A ⊆ On ∧ (x ∈ A ∧ (A ∩ x) = ∅)) → x ⊆ ∩A)
21		intss1 1979	. . . . . . . . . . 11 ⊢ (x ∈ A → ∩A ⊆ x)
22	21	ad2antrl 322	. . . . . . . . . 10 ⊢ ((A ⊆ On ∧ (x ∈ A ∧ (A ∩ x) = ∅)) → ∩A ⊆ x)
23	20, 22	eqssd 1518	. . . . . . . . 9 ⊢ ((A ⊆ On ∧ (x ∈ A ∧ (A ∩ x) = ∅)) → x = ∩A)
24	23	eleq1d 1155	. . . . . . . 8 ⊢ ((A ⊆ On ∧ (x ∈ A ∧ (A ∩ x) = ∅)) → (x ∈ A ↔ ∩A ∈ A))
25	24	biimpd 135	. . . . . . 7 ⊢ ((A ⊆ On ∧ (x ∈ A ∧ (A ∩ x) = ∅)) → (x ∈ A → ∩A ∈ A))
26	25	exp32 294	. . . . . 6 ⊢ (A ⊆ On → (x ∈ A → ((A ∩ x) = ∅ → (x ∈ A → ∩A ∈ A))))
27	26	com34 36	. . . . 5 ⊢ (A ⊆ On → (x ∈ A → (x ∈ A → ((A ∩ x) = ∅ → ∩A ∈ A))))
28	27	pm2.43d 59	. . . 4 ⊢ (A ⊆ On → (x ∈ A → ((A ∩ x) = ∅ → ∩A ∈ A)))
29	28	r19.23adv 1286	. . 3 ⊢ (A ⊆ On → (∃x ∈ A (A ∩ x) = ∅ → ∩A ∈ A))
30		ordon 2238	. . . 4 ⊢ Ord On
31		tz7.5 2220	. . . 4 ⊢ ((Ord On ∧ (A ⊆ On ∧ ¬ A = ∅)) → ∃x ∈ A (A ∩ x) = ∅)
32	30, 31	mpan 518	. . 3 ⊢ ((A ⊆ On ∧ ¬ A = ∅) → ∃x ∈ A (A ∩ x) = ∅)
33	29, 32	syl5 22	. 2 ⊢ (A ⊆ On → ((A ⊆ On ∧ ¬ A = ∅) → ∩A ∈ A))
34	33	anabsi5 377	1 ⊢ ((A ⊆ On ∧ ¬ A = ∅) → ∩A ∈ A)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707  ∩cint 1965  Ord word 2198  Oncon0 2199

This theorem is referenced by:  onint0 2262  onssmin 2263  onminsb 2264  onminesb 2265  oninton 2267  oneqmin 2273  onminex 2275  unblem1 3431  unblem2 3432  tz9.12lem3 3505  rankr1 3518  scott0 3542  oncardid 3628  cardid 3635  cardcf 3706

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-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  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-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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