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Theorem trelpss 27627
Description: An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 4558, ax-reg 7552 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 7562 . . 3  |-  _E  Fr  A
2 tz7.2 4558 . . 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 3328 . 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 1725    =/= wne 2598    C_ wss 3312    C. wpss 3313   Tr wtr 4294    _E cep 4484    Fr wfr 4530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-reg 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-fr 4533
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