Theorem tarax3c 15224

Description: 3rd axiom of a Tarski's class. A subset of a Tarski's class non equipotent to the class is an element of the class.

Assertion

Ref	Expression
tarax3c	|- ((T e. Tarski /\ A C_ T /\ -. A ~~ T) -> A e. T)

Proof of Theorem tarax3c

Step	Hyp	Ref	Expression
1		ssexg 3457	. . . . . 6 |- ((A C_ T /\ T e. Tarski ) -> A e. _V)
2	1	ancoms 484	. . . . 5 |- ((T e. Tarski /\ A C_ T) -> A e. _V)
3	2	3adant3 896	. . . 4 |- ((T e. Tarski /\ A C_ T /\ -. A ~~ T) -> A e. _V)
4		elpwg 3038	. . . . . 6 |- (A e. _V -> (A e. ~PT <-> A C_ T))
5	4	bicomd 580	. . . . 5 |- (A e. _V -> (A C_ T <-> A e. ~PT))
6	5	3anbi2d 1173	. . . 4 |- (A e. _V -> ((T e. Tarski /\ A C_ T /\ -. A ~~ T) <-> (T e. Tarski /\ A e. ~PT /\ -. A ~~ T)))
7	3, 6	syl 12	. . 3 |- ((T e. Tarski /\ A C_ T /\ -. A ~~ T) -> ((T e. Tarski /\ A C_ T /\ -. A ~~ T) <-> (T e. Tarski /\ A e. ~PT /\ -. A ~~ T)))
8		tarax3b 15223	. . 3 |- ((T e. Tarski /\ A e. ~PT /\ -. A ~~ T) -> A e. T)
9	7, 8	syl6bi 231	. 2 |- ((T e. Tarski /\ A C_ T /\ -. A ~~ T) -> ((T e. Tarski /\ A C_ T /\ -. A ~~ T) -> A e. T))
10	9	pm2.43i 78	1 |- ((T e. Tarski /\ A C_ T /\ -. A ~~ T) -> A e. T)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ w3a 858   e. wcel 1300  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032   class class class wbr 3338   ~~ cen 5423   Tarski ctarski 15208

This theorem is referenced by:  tarsuc2 15245

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-un 2600  df-in 2603  df-ss 2605  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-tsk 15210

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