Theorem inttaror 15277

Description: The intersection of a Tarski's class with the class of the ordinal numbers is an ordinal number.

Assertion

Ref	Expression
inttaror	|- (T e. Tarski -> (On i^i T) e. On)

Proof of Theorem inttaror

Step	Hyp	Ref	Expression
1		inex1g 3454	. . 3 |- (T e. Tarski -> (T i^i On) e. _V)
2		incom 2787	. . 3 |- (T i^i On) = (On i^i T)
3	1, 2	syl5eqelr 1976	. 2 |- (T e. Tarski -> (On i^i T) e. _V)
4		inss1 2812	. . . . . . 7 |- (On i^i T) C_ On
5	4	sseli 2617	. . . . . 6 |- (x e. (On i^i T) -> x e. On)
6	5	adantl 424	. . . . 5 |- ((T e. Tarski /\ x e. (On i^i T)) -> x e. On)
7		elin 2786	. . . . . . 7 |- (x e. (On i^i T) <-> (x e. On /\ x e. T))
8		eloni 3667	. . . . . . . . . 10 |- (x e. On -> Ord x)
9		ordtr 3672	. . . . . . . . . 10 |- (Ord x -> Tr x)
10		onss 3869	. . . . . . . . . . . 12 |- (x e. On -> x C_ On)
11		ssin 2814	. . . . . . . . . . . . . . . . 17 |- ((x C_ On /\ x C_ T) <-> x C_ (On i^i T))
12	11	biimpi 168	. . . . . . . . . . . . . . . 16 |- ((x C_ On /\ x C_ T) -> x C_ (On i^i T))
13	12	ex 402	. . . . . . . . . . . . . . 15 |- (x C_ On -> (x C_ T -> x C_ (On i^i T)))
14		tartrel 15239	. . . . . . . . . . . . . . 15 |- ((T e. Tarski /\ Tr x /\ x e. T) -> x C_ T)
15	13, 14	syl5com 63	. . . . . . . . . . . . . 14 |- ((T e. Tarski /\ Tr x /\ x e. T) -> (x C_ On -> x C_ (On i^i T)))
16	15	3exp 1066	. . . . . . . . . . . . 13 |- (T e. Tarski -> (Tr x -> (x e. T -> (x C_ On -> x C_ (On i^i T)))))
17	16	com14 42	. . . . . . . . . . . 12 |- (x C_ On -> (Tr x -> (x e. T -> (T e. Tarski -> x C_ (On i^i T)))))
18	10, 17	syl 12	. . . . . . . . . . 11 |- (x e. On -> (Tr x -> (x e. T -> (T e. Tarski -> x C_ (On i^i T)))))
19	18	com12 14	. . . . . . . . . 10 |- (Tr x -> (x e. On -> (x e. T -> (T e. Tarski -> x C_ (On i^i T)))))
20	8, 9, 19	3syl 24	. . . . . . . . 9 |- (x e. On -> (x e. On -> (x e. T -> (T e. Tarski -> x C_ (On i^i T)))))
21	20	pm2.43i 78	. . . . . . . 8 |- (x e. On -> (x e. T -> (T e. Tarski -> x C_ (On i^i T))))
22	21	imp 377	. . . . . . 7 |- ((x e. On /\ x e. T) -> (T e. Tarski -> x C_ (On i^i T)))
23	7, 22	sylbi 216	. . . . . 6 |- (x e. (On i^i T) -> (T e. Tarski -> x C_ (On i^i T)))
24	23	impcom 378	. . . . 5 |- ((T e. Tarski /\ x e. (On i^i T)) -> x C_ (On i^i T))
25	6, 24	jca 310	. . . 4 |- ((T e. Tarski /\ x e. (On i^i T)) -> (x e. On /\ x C_ (On i^i T)))
26	25	ex 402	. . 3 |- (T e. Tarski -> (x e. (On i^i T) -> (x e. On /\ x C_ (On i^i T))))
27	26	r19.21aiv 2175	. 2 |- (T e. Tarski -> A.x e. (On i^i T)(x e. On /\ x C_ (On i^i T)))
28		celsor 14443	. 2 |- (((On i^i T) e. _V /\ A.x e. (On i^i T)(x e. On /\ x C_ (On i^i T))) -> (On i^i T) e. On)
29	3, 27, 28	syl11anc 524	1 |- (T e. Tarski -> (On i^i T) e. On)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   e. wcel 1300  A.wral 2105  _Vcvv 2292   i^i cin 2592   C_ wss 2593  Tr wtr 3411  Ord word 3656  Oncon0 3657   Tarski ctarski 15208

This theorem is referenced by:  inttarcar 15278  carinttar 15279  carinttar2 15280  cartarlim 15282

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  ax-reg 5695

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-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  df-tsk 15210

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