Theorem onint 3876

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 C_ On /\ A =/= (/)) -> |^|A e. A)

Proof of Theorem onint

Step	Hyp	Ref	Expression
1		ssel 2615	. . . . . . . . . . . . . . . . . . . 20 |- (A C_ On -> (z e. A -> z e. On))
2		ontri1 3695	. . . . . . . . . . . . . . . . . . . . . 22 |- ((x e. On /\ z e. On) -> (x C_ z <-> -. z e. x))
3		ssel 2615	. . . . . . . . . . . . . . . . . . . . . 22 |- (x C_ z -> (y e. x -> y e. z))
4	2, 3	syl6bir 232	. . . . . . . . . . . . . . . . . . . . 21 |- ((x e. On /\ z e. On) -> (-. z e. x -> (y e. x -> y e. z)))
5	4	ex 402	. . . . . . . . . . . . . . . . . . . 20 |- (x e. On -> (z e. On -> (-. z e. x -> (y e. x -> y e. z))))
6	1, 5	sylan9 517	. . . . . . . . . . . . . . . . . . 19 |- ((A C_ On /\ x e. On) -> (z e. A -> (-. z e. x -> (y e. x -> y e. z))))
7	6	com4r 45	. . . . . . . . . . . . . . . . . 18 |- (y e. x -> ((A C_ On /\ x e. On) -> (z e. A -> (-. z e. x -> y e. z))))
8	7	imp31 389	. . . . . . . . . . . . . . . . 17 |- (((y e. x /\ (A C_ On /\ x e. On)) /\ z e. A) -> (-. z e. x -> y e. z))
9	8	ralimdvaa 2171	. . . . . . . . . . . . . . . 16 |- ((y e. x /\ (A C_ On /\ x e. On)) -> (A.z e. A -. z e. x -> A.z e. A y e. z))
10		disj 2914	. . . . . . . . . . . . . . . 16 |- ((A i^i x) = (/) <-> A.z e. A -. z e. x)
11		visset 2295	. . . . . . . . . . . . . . . . 17 |- y e. _V
12	11	elint2 3221	. . . . . . . . . . . . . . . 16 |- (y e. |^|A <-> A.z e. A y e. z)
13	9, 10, 12	3imtr4g 612	. . . . . . . . . . . . . . 15 |- ((y e. x /\ (A C_ On /\ x e. On)) -> ((A i^i x) = (/) -> y e. |^|A))
14		ssel 2615	. . . . . . . . . . . . . . . 16 |- (A C_ On -> (x e. A -> x e. On))
15	14	imdistani 491	. . . . . . . . . . . . . . 15 |- ((A C_ On /\ x e. A) -> (A C_ On /\ x e. On))
16	13, 15	sylan2 500	. . . . . . . . . . . . . 14 |- ((y e. x /\ (A C_ On /\ x e. A)) -> ((A i^i x) = (/) -> y e. |^|A))
17	16	exp32 408	. . . . . . . . . . . . 13 |- (y e. x -> (A C_ On -> (x e. A -> ((A i^i x) = (/) -> y e. |^|A))))
18	17	com4l 43	. . . . . . . . . . . 12 |- (A C_ On -> (x e. A -> ((A i^i x) = (/) -> (y e. x -> y e. |^|A))))
19	18	imp32 390	. . . . . . . . . . 11 |- ((A C_ On /\ (x e. A /\ (A i^i x) = (/))) -> (y e. x -> y e. |^|A))
20	19	ssrdv 2622	. . . . . . . . . 10 |- ((A C_ On /\ (x e. A /\ (A i^i x) = (/))) -> x C_ |^|A)
21		intss1 3231	. . . . . . . . . . 11 |- (x e. A -> |^|A C_ x)
22	21	ad2antrl 442	. . . . . . . . . 10 |- ((A C_ On /\ (x e. A /\ (A i^i x) = (/))) -> |^|A C_ x)
23	20, 22	eqssd 2633	. . . . . . . . 9 |- ((A C_ On /\ (x e. A /\ (A i^i x) = (/))) -> x = |^|A)
24	23	eleq1d 1963	. . . . . . . 8 |- ((A C_ On /\ (x e. A /\ (A i^i x) = (/))) -> (x e. A <-> |^|A e. A))
25	24	biimpd 170	. . . . . . 7 |- ((A C_ On /\ (x e. A /\ (A i^i x) = (/))) -> (x e. A -> |^|A e. A))
26	25	exp32 408	. . . . . 6 |- (A C_ On -> (x e. A -> ((A i^i x) = (/) -> (x e. A -> |^|A e. A))))
27	26	com34 40	. . . . 5 |- (A C_ On -> (x e. A -> (x e. A -> ((A i^i x) = (/) -> |^|A e. A))))
28	27	pm2.43d 79	. . . 4 |- (A C_ On -> (x e. A -> ((A i^i x) = (/) -> |^|A e. A)))
29	28	r19.23adv 2215	. . 3 |- (A C_ On -> (E.x e. A (A i^i x) = (/) -> |^|A e. A))
30		ordon 3863	. . . 4 |- Ord On
31		tz7.5 3679	. . . 4 |- ((Ord On /\ A C_ On /\ A =/= (/)) -> E.x e. A (A i^i x) = (/))
32	30, 31	mp3an1 1178	. . 3 |- ((A C_ On /\ A =/= (/)) -> E.x e. A (A i^i x) = (/))
33	29, 32	syl5 20	. 2 |- (A C_ On -> ((A C_ On /\ A =/= (/)) -> |^|A e. A))
34	33	anabsi5 553	1 |- ((A C_ On /\ A =/= (/)) -> |^|A e. A)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   i^i cin 2592   C_ wss 2593  (/)c0 2875  |^|cint 3214  Ord word 3656  Oncon0 3657

This theorem is referenced by:  onint0 3877  onssmin 3878  onminsb 3879  onminesb 3880  oninton 3881  oneqmin 3886  onminex 3888  unblem1 5633  unblem2 5634  tz9.12lem3 5772  rankr1 5785  scott0 5847  oncardid 5867  cardid 5977  cardcf 6059  sltval2 13997  nocvxmin 14029  axfelem4 14034

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661

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