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Theorem tz7.2 4270
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 4019 . . 3  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
2 efrirr 4267 . . . . 5  |-  (  _E  Fr  A  ->  -.  A  e.  A )
3 eleq1 2313 . . . . . 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 2460 . . 3  |-  (  _E  Fr  A  ->  ( B  e.  A  ->  B  =/=  A ) )
71, 6anim12ii 556 . 2  |-  ( ( Tr  A  /\  _E  Fr  A )  ->  ( B  e.  A  ->  ( B  C_  A  /\  B  =/=  A ) ) )
873impia 1153 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 939    = wceq 1619    e. wcel 1621    =/= wne 2412    C_ wss 3078   Tr wtr 4010    _E cep 4196    Fr wfr 4242
This theorem is referenced by:  tz7.7  4311  trelpss  26827
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-fr 4245
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