Theorem onint 2996

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 e. A)

Proof of Theorem onint

Step	Hyp	Ref	Expression
1		ssel 2053	. . . . . . . . . . . . . . . . . . . 20 |- (A (_ On -> (z e. A -> z e. On))
2		ontri1 2971	. . . . . . . . . . . . . . . . . . . . . 22 |- ((x e. On /\ z e. On) -> (x (_ z <-> -. z e. x))
3		ssel 2053	. . . . . . . . . . . . . . . . . . . . . 22 |- (x (_ z -> (y e. x -> y e. z))
4	2, 3	syl6bir 215	. . . . . . . . . . . . . . . . . . . . 21 |- ((x e. On /\ z e. On) -> (-. z e. x -> (y e. x -> y e. z)))
5	4	ex 373	. . . . . . . . . . . . . . . . . . . 20 |- (x e. On -> (z e. On -> (-. z e. x -> (y e. x -> y e. z))))
6	1, 5	sylan9 468	. . . . . . . . . . . . . . . . . . 19 |- ((A (_ On /\ x e. On) -> (z e. A -> (-. z e. x -> (y e. x -> y e. z))))
7	6	com4r 41	. . . . . . . . . . . . . . . . . 18 |- (y e. x -> ((A (_ On /\ x e. On) -> (z e. A -> (-. z e. x -> y e. z))))
8	7	imp31 362	. . . . . . . . . . . . . . . . 17 |- (((y e. x /\ (A (_ On /\ x e. On)) /\ z e. A) -> (-. z e. x -> y e. z))
9	8	r19.20dva 1701	. . . . . . . . . . . . . . . 16 |- ((y e. x /\ (A (_ On /\ x e. On)) -> (A.z e. A -. z e. x -> A.z e. A y e. z))
10		disj 2301	. . . . . . . . . . . . . . . 16 |- ((A i^i x) = (/) <-> A.z e. A -. z e. x)
11		visset 1804	. . . . . . . . . . . . . . . . 17 |- y e. V
12	11	elint2 2530	. . . . . . . . . . . . . . . 16 |- (y e. |^|A <-> A.z e. A y e. z)
13	9, 10, 12	3imtr4g 551	. . . . . . . . . . . . . . 15 |- ((y e. x /\ (A (_ On /\ x e. On)) -> ((A i^i x) = (/) -> y e. |^|A))
14		ssel 2053	. . . . . . . . . . . . . . . 16 |- (A (_ On -> (x e. A -> x e. On))
15	14	imdistani 443	. . . . . . . . . . . . . . 15 |- ((A (_ On /\ x e. A) -> (A (_ On /\ x e. On))
16	13, 15	sylan2 451	. . . . . . . . . . . . . 14 |- ((y e. x /\ (A (_ On /\ x e. A)) -> ((A i^i x) = (/) -> y e. |^|A))
17	16	exp32 377	. . . . . . . . . . . . 13 |- (y e. x -> (A (_ On -> (x e. A -> ((A i^i x) = (/) -> y e. |^|A))))
18	17	com4l 39	. . . . . . . . . . . 12 |- (A (_ On -> (x e. A -> ((A i^i x) = (/) -> (y e. x -> y e. |^|A))))
19	18	imp32 363	. . . . . . . . . . 11 |- ((A (_ On /\ (x e. A /\ (A i^i x) = (/))) -> (y e. x -> y e. |^|A))
20	19	ssrdv 2060	. . . . . . . . . 10 |- ((A (_ On /\ (x e. A /\ (A i^i x) = (/))) -> x (_ |^|A)
21		intss1 2538	. . . . . . . . . . 11 |- (x e. A -> |^|A (_ x)
22	21	ad2antrl 406	. . . . . . . . . 10 |- ((A (_ On /\ (x e. A /\ (A i^i x) = (/))) -> |^|A (_ x)
23	20, 22	eqssd 2069	. . . . . . . . 9 |- ((A (_ On /\ (x e. A /\ (A i^i x) = (/))) -> x = |^|A)
24	23	eleq1d 1532	. . . . . . . 8 |- ((A (_ On /\ (x e. A /\ (A i^i x) = (/))) -> (x e. A <-> |^|A e. A))
25	24	biimpd 153	. . . . . . 7 |- ((A (_ On /\ (x e. A /\ (A i^i x) = (/))) -> (x e. A -> |^|A e. A))
26	25	exp32 377	. . . . . 6 |- (A (_ On -> (x e. A -> ((A i^i x) = (/) -> (x e. A -> |^|A e. A))))
27	26	com34 36	. . . . 5 |- (A (_ On -> (x e. A -> (x e. A -> ((A i^i x) = (/) -> |^|A e. A))))
28	27	pm2.43d 65	. . . 4 |- (A (_ On -> (x e. A -> ((A i^i x) = (/) -> |^|A e. A)))
29	28	r19.23adv 1738	. . 3 |- (A (_ On -> (E.x e. A (A i^i x) = (/) -> |^|A e. A))
30		ordon 2977	. . . 4 |- Ord On
31		tz7.5 2959	. . . 4 |- ((Ord On /\ A (_ On /\ A =/= (/)) -> E.x e. A (A i^i x) = (/))
32	30, 31	mp3an1 900	. . 3 |- ((A (_ On /\ A =/= (/)) -> E.x e. A (A i^i x) = (/))
33	29, 32	syl5 21	. 2 |- (A (_ On -> ((A (_ On /\ A =/= (/)) -> |^|A e. A))
34	33	anabsi5 494	1 |- ((A (_ On /\ A =/= (/)) -> |^|A e. A)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577  A.wral 1637  E.wrex 1638   i^i cin 2036   (_ wss 2037  (/)c0 2270  |^|cint 2523  Ord word 2937  Oncon0 2938

This theorem is referenced by:  onint0 2997  onssmin 2998  onminsb 2999  onminesb 3000  oninton 3002  oneqmin 3008  onminex 3010  unblem1 4517  unblem2 4518  tz9.12lem3 4633  rankr1 4646  scott0 4689  oncardid 4793  cardid 4800  cardcf 4883

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942

Copyright terms: Public domain