Theorem carinttar 15279

Description: The cardinal of the intersection of a Tarski's class with the class of the ordinal numbers.

Assertion

Ref	Expression
carinttar	|- (T e. Tarski -> (card` (On i^i T)) = (On i^i T))

Proof of Theorem carinttar

Step	Hyp	Ref	Expression
1		iscard 6005	. 2 |- ((card` (On i^i T)) = (On i^i T) <-> ((On i^i T) e. On /\ A.x e. (On i^i T)x ~< (On i^i T)))
2		inttaror 15277	. 2 |- (T e. Tarski -> (On i^i T) e. On)
3		brsdom 5440	. . . . 5 |- (x ~< (On i^i T) <-> (x ~<_ (On i^i T) /\ -. x ~~ (On i^i T)))
4		eltintpar 15276	. . . . . . 7 |- (T e. Tarski -> (x e. (On i^i T) -> x C_ (On i^i T)))
5		inex1g 3454	. . . . . . . . 9 |- (T e. Tarski -> (T i^i On) e. _V)
6		incom 2787	. . . . . . . . 9 |- (On i^i T) = (T i^i On)
7	5, 6	syl5eqel 1975	. . . . . . . 8 |- (T e. Tarski -> (On i^i T) e. _V)
8		ssdom2g 5468	. . . . . . . 8 |- ((On i^i T) e. _V -> (x C_ (On i^i T) -> x ~<_ (On i^i T)))
9	7, 8	syl 12	. . . . . . 7 |- (T e. Tarski -> (x C_ (On i^i T) -> x ~<_ (On i^i T)))
10	4, 9	syld 30	. . . . . 6 |- (T e. Tarski -> (x e. (On i^i T) -> x ~<_ (On i^i T)))
11	10	imp 377	. . . . 5 |- ((T e. Tarski /\ x e. (On i^i T)) -> x ~<_ (On i^i T))
12		inttarcar 15278	. . . . . . 7 |- (T e. Tarski -> (On i^i T) ~~ T)
13		elin 2786	. . . . . . . . 9 |- (x e. (On i^i T) <-> (x e. On /\ x e. T))
14		tarax3d2 15225	. . . . . . . . . . . 12 |- ((T e. Tarski /\ x e. T) -> x ~< T)
15		sdomnen 5446	. . . . . . . . . . . . 13 |- (x ~< T -> -. x ~~ T)
16		entr 5473	. . . . . . . . . . . . . . . . 17 |- ((x ~~ (On i^i T) /\ (On i^i T) ~~ T) -> x ~~ T)
17	16	expcom 403	. . . . . . . . . . . . . . . 16 |- ((On i^i T) ~~ T -> (x ~~ (On i^i T) -> x ~~ T))
18	17	con3d 111	. . . . . . . . . . . . . . 15 |- ((On i^i T) ~~ T -> (-. x ~~ T -> -. x ~~ (On i^i T)))
19	18	a1i 8	. . . . . . . . . . . . . 14 |- ((T e. Tarski /\ x e. T) -> ((On i^i T) ~~ T -> (-. x ~~ T -> -. x ~~ (On i^i T))))
20	19	com3r 39	. . . . . . . . . . . . 13 |- (-. x ~~ T -> ((T e. Tarski /\ x e. T) -> ((On i^i T) ~~ T -> -. x ~~ (On i^i T))))
21	15, 20	syl 12	. . . . . . . . . . . 12 |- (x ~< T -> ((T e. Tarski /\ x e. T) -> ((On i^i T) ~~ T -> -. x ~~ (On i^i T))))
22	14, 21	mpcom 60	. . . . . . . . . . 11 |- ((T e. Tarski /\ x e. T) -> ((On i^i T) ~~ T -> -. x ~~ (On i^i T)))
23	22	expcom 403	. . . . . . . . . 10 |- (x e. T -> (T e. Tarski -> ((On i^i T) ~~ T -> -. x ~~ (On i^i T))))
24	23	adantl 424	. . . . . . . . 9 |- ((x e. On /\ x e. T) -> (T e. Tarski -> ((On i^i T) ~~ T -> -. x ~~ (On i^i T))))
25	13, 24	sylbi 216	. . . . . . . 8 |- (x e. (On i^i T) -> (T e. Tarski -> ((On i^i T) ~~ T -> -. x ~~ (On i^i T))))
26	25	com13 37	. . . . . . 7 |- ((On i^i T) ~~ T -> (T e. Tarski -> (x e. (On i^i T) -> -. x ~~ (On i^i T))))
27	12, 26	mpcom 60	. . . . . 6 |- (T e. Tarski -> (x e. (On i^i T) -> -. x ~~ (On i^i T)))
28	27	imp 377	. . . . 5 |- ((T e. Tarski /\ x e. (On i^i T)) -> -. x ~~ (On i^i T))
29	3, 11, 28	sylanbrc 527	. . . 4 |- ((T e. Tarski /\ x e. (On i^i T)) -> x ~< (On i^i T))
30	29	ex 402	. . 3 |- (T e. Tarski -> (x e. (On i^i T) -> x ~< (On i^i T)))
31	30	r19.21aiv 2175	. 2 |- (T e. Tarski -> A.x e. (On i^i T)x ~< (On i^i T))
32	1, 2, 31	sylanbrc 527	1 |- (T e. Tarski -> (card` (On i^i T)) = (On i^i T))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   i^i cin 2592   C_ wss 2593   class class class wbr 3338  Oncon0 3657  ` cfv 3998   ~~ cen 5423   ~<_ cdom 5424   ~< csdm 5425  cardccrd 5859   Tarski ctarski 15208

This theorem is referenced by:  carinttar2 15280

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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-ac 5906

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-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  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-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-card 5862  df-tsk 15210

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