Theorem tartwo 15233

Description: Two is an element of a non empty Tarski's class.

Assertion

Ref	Expression
tartwo	|- ((T e. Tarski /\ T =/= (/)) -> 2o e. T)

Proof of Theorem tartwo

Step	Hyp	Ref	Expression
1		tarone 15232	. . 3 |- ((T e. Tarski /\ T =/= (/)) -> 1o e. T)
2		df1o2 5185	. . . . 5 |- 1o = {(/)}
3	2	eleq1i 1960	. . . 4 |- (1o e. T <-> {(/)} e. T)
4		tarax2 15217	. . . . . . . 8 |- ((T e. Tarski /\ {(/)} e. T) -> ~P{(/)} e. T)
5		pwpw0 3134	. . . . . . . 8 |- ~P{(/)} = {(/), {(/)}}
6	4, 5	syl5eqelr 1976	. . . . . . 7 |- ((T e. Tarski /\ {(/)} e. T) -> {(/), {(/)}} e. T)
7	6	ex 402	. . . . . 6 |- (T e. Tarski -> ({(/)} e. T -> {(/), {(/)}} e. T))
8	7	adantr 425	. . . . 5 |- ((T e. Tarski /\ T =/= (/)) -> ({(/)} e. T -> {(/), {(/)}} e. T))
9	8	com12 14	. . . 4 |- ({(/)} e. T -> ((T e. Tarski /\ T =/= (/)) -> {(/), {(/)}} e. T))
10	3, 9	sylbi 216	. . 3 |- (1o e. T -> ((T e. Tarski /\ T =/= (/)) -> {(/), {(/)}} e. T))
11	1, 10	mpcom 60	. 2 |- ((T e. Tarski /\ T =/= (/)) -> {(/), {(/)}} e. T)
12		df2o2 5186	. 2 |- 2o = {(/), {(/)}}
13	11, 12	syl5eqel 1975	1 |- ((T e. Tarski /\ T =/= (/)) -> 2o e. T)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300   =/= wne 2017  (/)c0 2875  ~Pcpw 3032  {csn 3044  {cpr 3045  1oc1o 5172  2oc2o 5173   Tarski ctarski 15208

This theorem is referenced by:  tarmrtwo 15234

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-nul 3445  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-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-suc 3663  df-1o 5177  df-2o 5178  df-tsk 15210

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