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Theorem tz7.2 4091
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 3863 . . 3  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
2 efrirr 4090 . . . . 5  |-  (  _E  Fr  A  ->  -.  A  e.  A )
3 eleq1 2100 . . . . . 6  |-  ( B  =  A  ->  ( B  e.  A  <->  A  e.  A ) )
43notbid 592 . . . . 5  |-  ( B  =  A  ->  ( -.  B  e.  A  <->  -.  A  e.  A ) )
52, 4syl5ibrcom 146 . . . 4  |-  (  _E  Fr  A  ->  ( B  =  A  ->  -.  B  e.  A ) )
65necon2ad 2262 . . 3  |-  (  _E  Fr  A  ->  ( B  e.  A  ->  B  =/=  A ) )
71, 6anim12ii 325 . 2  |-  ( ( Tr  A  /\  _E  Fr  A )  ->  ( B  e.  A  ->  ( B  C_  A  /\  B  =/=  A ) ) )
873impia 1101 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 97    /\ w3a 885    = wceq 1243    e. wcel 1393    =/= wne 2204    C_ wss 2917   Tr wtr 3854    _E cep 4024    Fr wfr 4065
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-tr 3855  df-eprel 4026  df-frfor 4068  df-frind 4069
This theorem is referenced by: (None)
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