Theorem carinttar2 15280

Description: The cardinal of a Tarski's class equals the intersection of the Tarski's class with the class of the ordinal numbers. CLASSES2 th. 10.

Assertion

Ref	Expression
carinttar2	|- (T e. Tarski -> (card` T) = (On i^i T))

Proof of Theorem carinttar2

Step	Hyp	Ref	Expression
1		inttarcar 15278	. . . 4 |- (T e. Tarski -> (On i^i T) ~~ T)
2		ensymg 5470	. . . 4 |- (T e. Tarski -> ((On i^i T) ~~ T -> T ~~ (On i^i T)))
3	1, 2	mpd 29	. . 3 |- (T e. Tarski -> T ~~ (On i^i T))
4		inttaror 15277	. . . 4 |- (T e. Tarski -> (On i^i T) e. On)
5		carden 5981	. . . 4 |- ((T e. Tarski /\ (On i^i T) e. On) -> ((card` T) = (card` (On i^i T)) <-> T ~~ (On i^i T)))
6	4, 5	mpdan 768	. . 3 |- (T e. Tarski -> ((card` T) = (card` (On i^i T)) <-> T ~~ (On i^i T)))
7	3, 6	mpbird 213	. 2 |- (T e. Tarski -> (card` T) = (card` (On i^i T)))
8		carinttar 15279	. 2 |- (T e. Tarski -> (card` (On i^i T)) = (On i^i T))
9	7, 8	eqtrd 1925	1 |- (T e. Tarski -> (card` T) = (On i^i T))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300   i^i cin 2592   class class class wbr 3338  Oncon0 3657  ` cfv 3998   ~~ cen 5423  cardccrd 5859   Tarski ctarski 15208

This theorem is referenced by:  cardtar 15281  cartarlim 15282  elcarelcl 15283

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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