Theorem tarax2 15217

Description: 2nd axiom of a a Tarski's class. The powerset of an element of a Tarski's class belongs to the class.

Assertion

Ref	Expression
tarax2	|- ((T e. Tarski /\ A e. T) -> ~PA e. T)

Proof of Theorem tarax2

Step	Hyp	Ref	Expression
1		pweq 3036	. . . . 5 |- (x = A -> ~Px = ~PA)
2	1	eleq1d 1963	. . . 4 |- (x = A -> (~Px e. T <-> ~PA e. T))
3	2	imbi2d 674	. . 3 |- (x = A -> ((T e. Tarski -> ~Px e. T) <-> (T e. Tarski -> ~PA e. T)))
4		tarval1g 15215	. . . . . 6 |- (T e. Tarski -> (T e. Tarski <-> (A.x e. T (~Px C_ T /\ ~Px e. T) /\ A.x e. ~P T(x ~~ T \/ x e. T))))
5		simpr 350	. . . . . . . 8 |- ((~Px C_ T /\ ~Px e. T) -> ~Px e. T)
6	5	ralimi 2168	. . . . . . 7 |- (A.x e. T (~Px C_ T /\ ~Px e. T) -> A.x e. T ~Px e. T)
7	6	adantr 425	. . . . . 6 |- ((A.x e. T (~Px C_ T /\ ~Px e. T) /\ A.x e. ~P T(x ~~ T \/ x e. T)) -> A.x e. T ~Px e. T)
8	4, 7	syl6bi 231	. . . . 5 |- (T e. Tarski -> (T e. Tarski -> A.x e. T ~Px e. T))
9	8	pm2.43i 78	. . . 4 |- (T e. Tarski -> A.x e. T ~Px e. T)
10	9	r19.21be 2191	. . 3 |- A.x e. T (T e. Tarski -> ~Px e. T)
11	3, 10	vtoclri 2360	. 2 |- (A e. T -> (T e. Tarski -> ~PA e. T))
12	11	impcom 378	1 |- ((T e. Tarski /\ A e. T) -> ~PA e. T)

Syntax hints:   -> wi 3   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  ~Pcpw 3032   class class class wbr 3338   ~~ cen 5423   Tarski ctarski 15208

This theorem is referenced by:  tarsin 15230  tartwo 15233  tclinf 15241  sexptrt 15243  intartar 15255  pwtsm 15266

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-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-pow 3481

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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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