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Theorem tz7.2 4378
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 4125 . . 3  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
2 efrirr 4375 . . . . 5  |-  (  _E  Fr  A  ->  -.  A  e.  A )
3 eleq1 2346 . . . . . 6  |-  ( B  =  A  ->  ( B  e.  A  <->  A  e.  A ) )
43notbid 287 . . . . 5  |-  ( B  =  A  ->  ( -.  B  e.  A  <->  -.  A  e.  A ) )
52, 4syl5ibrcom 215 . . . 4  |-  (  _E  Fr  A  ->  ( B  =  A  ->  -.  B  e.  A ) )
65necon2ad 2497 . . 3  |-  (  _E  Fr  A  ->  ( B  e.  A  ->  B  =/=  A ) )
71, 6anim12ii 555 . 2  |-  ( ( Tr  A  /\  _E  Fr  A )  ->  ( B  e.  A  ->  ( B  C_  A  /\  B  =/=  A ) ) )
873impia 1150 1  |-  ( ( Tr  A  /\  _E  Fr  A  /\  B  e.  A )  ->  ( B  C_  A  /\  B  =/=  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1625    e. wcel 1687    =/= wne 2449    C_ wss 3155   Tr wtr 4116    _E cep 4304    Fr wfr 4350
This theorem is referenced by:  tz7.7  4419  trelpss  27061
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pr 4215
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-ral 2551  df-rex 2552  df-rab 2555  df-v 2793  df-sbc 2995  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3831  df-br 4027  df-opab 4081  df-tr 4117  df-eprel 4306  df-fr 4353
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