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Theorem trelpss 27763
Description: An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 4393, ax-reg 7322 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.)
Assertion
Ref Expression
trelpss  |-  ( ( Tr  A  /\  B  e.  A )  ->  B  C.  A )

Proof of Theorem trelpss
StepHypRef Expression
1 zfregfr 7332 . . 3  |-  _E  Fr  A
2 tz7.2 4393 . . 3  |-  ( ( Tr  A  /\  _E  Fr  A  /\  B  e.  A )  ->  ( B  C_  A  /\  B  =/=  A ) )
31, 2mp3an2 1265 . 2  |-  ( ( Tr  A  /\  B  e.  A )  ->  ( B  C_  A  /\  B  =/=  A ) )
4 df-pss 3181 . 2  |-  ( B 
C.  A  <->  ( B  C_  A  /\  B  =/= 
A ) )
53, 4sylibr 203 1  |-  ( ( Tr  A  /\  B  e.  A )  ->  B  C.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696    =/= wne 2459    C_ wss 3165    C. wpss 3166   Tr wtr 4129    _E cep 4319    Fr wfr 4365
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-reg 7322
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-fr 4368
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