Theorem emptar 15231

Description: A non empty Tarski's class contains the empty set.

Assertion

Ref	Expression
emptar	|- ((T e. Tarski /\ T =/= (/)) -> (/) e. T)

Proof of Theorem emptar

Step	Hyp	Ref	Expression
1		n0 2884	. . . 4 |- (T =/= (/) <-> E.x x e. T)
2	1	biimpi 168	. . 3 |- (T =/= (/) -> E.x x e. T)
3	2	adantl 424	. 2 |- ((T e. Tarski /\ T =/= (/)) -> E.x x e. T)
4		tarax1 15216	. . . . . . 7 |- ((T e. Tarski /\ x e. T) -> ~Px C_ T)
5		0elpw 3473	. . . . . . . 8 |- (/) e. ~Px
6		ssel 2615	. . . . . . . 8 |- (~Px C_ T -> ((/) e. ~Px -> (/) e. T))
7	5, 6	mpi 55	. . . . . . 7 |- (~Px C_ T -> (/) e. T)
8	4, 7	syl 12	. . . . . 6 |- ((T e. Tarski /\ x e. T) -> (/) e. T)
9	8	ex 402	. . . . 5 |- (T e. Tarski -> (x e. T -> (/) e. T))
10	9	adantr 425	. . . 4 |- ((T e. Tarski /\ T =/= (/)) -> (x e. T -> (/) e. T))
11	10	com12 14	. . 3 |- (x e. T -> ((T e. Tarski /\ T =/= (/)) -> (/) e. T))
12	11	19.23aiv 1674	. 2 |- (E.x x e. T -> ((T e. Tarski /\ T =/= (/)) -> (/) e. T))
13	3, 12	mpcom 60	1 |- ((T e. Tarski /\ T =/= (/)) -> (/) e. T)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  E.wex 1326   =/= wne 2017   C_ wss 2593  (/)c0 2875  ~Pcpw 3032   Tarski ctarski 15208

This theorem is referenced by:  tarone 15232  cptarc 15242  cartarlim 15282

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-nul 3445

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-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-tsk 15210

Copyright terms: Public domain