Theorem eltintpar 15276

Description: An element of the intersection of a Tarski's class with the class of the ordinal numbers is a part of the intersection.

Assertion

Ref	Expression
eltintpar	|- (T e. Tarski -> (A e. (On i^i T) -> A C_ (On i^i T)))

Proof of Theorem eltintpar

Step	Hyp	Ref	Expression
1		elin 2786	. . . . 5 |- (A e. (On i^i T) <-> (A e. On /\ A e. T))
2		onss 3869	. . . . . . . 8 |- (A e. On -> A C_ On)
3	2	ad2antrr 440	. . . . . . 7 |- (((A e. On /\ A e. T) /\ T e. Tarski ) -> A C_ On)
4		eloni 3667	. . . . . . . . 9 |- (A e. On -> Ord A)
5		ordtr 3672	. . . . . . . . 9 |- (Ord A -> Tr A)
6		tartrel 15239	. . . . . . . . . . 11 |- ((T e. Tarski /\ Tr A /\ A e. T) -> A C_ T)
7	6	3exp 1066	. . . . . . . . . 10 |- (T e. Tarski -> (Tr A -> (A e. T -> A C_ T)))
8	7	com3l 38	. . . . . . . . 9 |- (Tr A -> (A e. T -> (T e. Tarski -> A C_ T)))
9	4, 5, 8	3syl 24	. . . . . . . 8 |- (A e. On -> (A e. T -> (T e. Tarski -> A C_ T)))
10	9	imp31 389	. . . . . . 7 |- (((A e. On /\ A e. T) /\ T e. Tarski ) -> A C_ T)
11	3, 10	jca 310	. . . . . 6 |- (((A e. On /\ A e. T) /\ T e. Tarski ) -> (A C_ On /\ A C_ T))
12	11	ex 402	. . . . 5 |- ((A e. On /\ A e. T) -> (T e. Tarski -> (A C_ On /\ A C_ T)))
13	1, 12	sylbi 216	. . . 4 |- (A e. (On i^i T) -> (T e. Tarski -> (A C_ On /\ A C_ T)))
14	13	impcom 378	. . 3 |- ((T e. Tarski /\ A e. (On i^i T)) -> (A C_ On /\ A C_ T))
15		ssin 2814	. . 3 |- ((A C_ On /\ A C_ T) <-> A C_ (On i^i T))
16	14, 15	sylib 215	. 2 |- ((T e. Tarski /\ A e. (On i^i T)) -> A C_ (On i^i T))
17	16	ex 402	1 |- (T e. Tarski -> (A e. (On i^i T) -> A C_ (On i^i T)))

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

This theorem is referenced by:  carinttar 15279

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-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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