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Theorem tskpr 8408
Description: If  A and  B are members of a Tarski's class, their unordered pair is also an element of the class. JFM CLASSES2 th. 3 (partly). (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Jun-2013.)
Assertion
Ref Expression
tskpr  |-  ( ( T  e.  Tarski  /\  A  e.  T  /\  B  e.  T )  ->  { A ,  B }  e.  T
)

Proof of Theorem tskpr
StepHypRef Expression
1 simp1 955 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T  /\  B  e.  T )  ->  T  e.  Tarski )
2 prssi 3787 . . 3  |-  ( ( A  e.  T  /\  B  e.  T )  ->  { A ,  B }  C_  T )
323adant1 973 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T  /\  B  e.  T )  ->  { A ,  B }  C_  T
)
4 prfi 7147 . . . . 5  |-  { A ,  B }  e.  Fin
5 isfinite 7369 . . . . 5  |-  ( { A ,  B }  e.  Fin  <->  { A ,  B }  ~<  om )
64, 5mpbi 199 . . . 4  |-  { A ,  B }  ~<  om
7 ne0i 3474 . . . . 5  |-  ( A  e.  T  ->  T  =/=  (/) )
8 tskinf 8407 . . . . 5  |-  ( ( T  e.  Tarski  /\  T  =/=  (/) )  ->  om  ~<_  T )
97, 8sylan2 460 . . . 4  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  om  ~<_  T )
10 sdomdomtr 7010 . . . 4  |-  ( ( { A ,  B }  ~<  om  /\  om  ~<_  T )  ->  { A ,  B }  ~<  T )
116, 9, 10sylancr 644 . . 3  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  { A ,  B }  ~<  T )
12113adant3 975 . 2  |-  ( ( T  e.  Tarski  /\  A  e.  T  /\  B  e.  T )  ->  { A ,  B }  ~<  T )
13 tskssel 8395 . 2  |-  ( ( T  e.  Tarski  /\  { A ,  B }  C_  T  /\  { A ,  B }  ~<  T )  ->  { A ,  B }  e.  T
)
141, 3, 12, 13syl3anc 1182 1  |-  ( ( T  e.  Tarski  /\  A  e.  T  /\  B  e.  T )  ->  { A ,  B }  e.  T
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468   {cpr 3654   class class class wbr 4039   omcom 4672    ~<_ cdom 6877    ~< csdm 6878   Fincfn 6879   Tarskictsk 8386
This theorem is referenced by:  tskop  8409  tskwun  8422  tskun  8424  grutsk1  8459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-r1 7452  df-tsk 8387
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