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Theorem tz7.2 4414
Description: Similar to Theorem 7.2 of [TakeutiZaring] p. 35, of except that the Axiom of Regularity is not required due to antecedent  _E  Fr  A. (Contributed by NM, 4-May-1994.)
Assertion
Ref Expression
tz7.2  |-  ( ( Tr  A  /\  _E  Fr  A  /\  B  e.  A )  ->  ( B  C_  A  /\  B  =/=  A ) )

Proof of Theorem tz7.2
StepHypRef Expression
1 trss 4159 . . 3  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
2 efrirr 4411 . . . . 5  |-  (  _E  Fr  A  ->  -.  A  e.  A )
3 eleq1 2376 . . . . . 6  |-  ( B  =  A  ->  ( B  e.  A  <->  A  e.  A ) )
43notbid 285 . . . . 5  |-  ( B  =  A  ->  ( -.  B  e.  A  <->  -.  A  e.  A ) )
52, 4syl5ibrcom 213 . . . 4  |-  (  _E  Fr  A  ->  ( B  =  A  ->  -.  B  e.  A ) )
65necon2ad 2527 . . 3  |-  (  _E  Fr  A  ->  ( B  e.  A  ->  B  =/=  A ) )
71, 6anim12ii 553 . 2  |-  ( ( Tr  A  /\  _E  Fr  A )  ->  ( B  e.  A  ->  ( B  C_  A  /\  B  =/=  A ) ) )
873impia 1148 1  |-  ( ( Tr  A  /\  _E  Fr  A  /\  B  e.  A )  ->  ( B  C_  A  /\  B  =/=  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479    C_ wss 3186   Tr wtr 4150    _E cep 4340    Fr wfr 4386
This theorem is referenced by:  tz7.7  4455  trelpss  26808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-tr 4151  df-eprel 4342  df-fr 4389
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