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Theorem trelpss 27321
Description: An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 4500, ax-reg 7486 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 7496 . . 3  |-  _E  Fr  A
2 tz7.2 4500 . . 3  |-  ( ( Tr  A  /\  _E  Fr  A  /\  B  e.  A )  ->  ( B  C_  A  /\  B  =/=  A ) )
31, 2mp3an2 1267 . 2  |-  ( ( Tr  A  /\  B  e.  A )  ->  ( B  C_  A  /\  B  =/=  A ) )
4 df-pss 3272 . 2  |-  ( B 
C.  A  <->  ( B  C_  A  /\  B  =/= 
A ) )
53, 4sylibr 204 1  |-  ( ( Tr  A  /\  B  e.  A )  ->  B  C.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717    =/= wne 2543    C_ wss 3256    C. wpss 3257   Tr wtr 4236    _E cep 4426    Fr wfr 4472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-reg 7486
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-tr 4237  df-eprel 4428  df-fr 4475
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