Theorem inacint 15221

Description: The intersection of two elements of a Tarski's class belongs to the class.

Assertion

Ref	Expression
inacint	|- ((T e. Tarski /\ A e. T /\ B e. T) -> (A i^i B) e. T)

Proof of Theorem inacint

Step	Hyp	Ref	Expression
1		tarax1 15216	. . 3 |- ((T e. Tarski /\ A e. T) -> ~PA C_ T)
2	1	3adant3 896	. 2 |- ((T e. Tarski /\ A e. T /\ B e. T) -> ~PA C_ T)
3		ssid 2634	. . . . 5 |- A C_ A
4		elpwg 3038	. . . . 5 |- (A e. T -> (A e. ~PA <-> A C_ A))
5	3, 4	mpbiri 211	. . . 4 |- (A e. T -> A e. ~PA)
6	5	3ad2ant2 898	. . 3 |- ((T e. Tarski /\ A e. T /\ B e. T) -> A e. ~PA)
7		inpws1 14345	. . 3 |- (A e. ~PA -> (A i^i B) e. ~PA)
8	6, 7	syl 12	. 2 |- ((T e. Tarski /\ A e. T /\ B e. T) -> (A i^i B) e. ~PA)
9	2, 8	sseldd 2620	1 |- ((T e. Tarski /\ A e. T /\ B e. T) -> (A i^i B) e. T)

Syntax hints:   -> wi 3   /\ w3a 858   e. wcel 1300   i^i cin 2592   C_ wss 2593  ~Pcpw 3032   Tarski ctarski 15208

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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