MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz7.2 Structured version   Visualization version   Unicode version

Theorem tz7.2 4823
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 4499 . . 3  |-  ( Tr  A  ->  ( B  e.  A  ->  B  C_  A ) )
2 efrirr 4820 . . . . 5  |-  (  _E  Fr  A  ->  -.  A  e.  A )
3 eleq1 2537 . . . . . 6  |-  ( B  =  A  ->  ( B  e.  A  <->  A  e.  A ) )
43notbid 301 . . . . 5  |-  ( B  =  A  ->  ( -.  B  e.  A  <->  -.  A  e.  A ) )
52, 4syl5ibrcom 230 . . . 4  |-  (  _E  Fr  A  ->  ( B  =  A  ->  -.  B  e.  A ) )
65necon2ad 2658 . . 3  |-  (  _E  Fr  A  ->  ( B  e.  A  ->  B  =/=  A ) )
71, 6anim12ii 580 . 2  |-  ( ( Tr  A  /\  _E  Fr  A )  ->  ( B  e.  A  ->  ( B  C_  A  /\  B  =/=  A ) ) )
873impia 1228 1  |-  ( ( Tr  A  /\  _E  Fr  A  /\  B  e.  A )  ->  ( B  C_  A  /\  B  =/=  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641    C_ wss 3390   Tr wtr 4490    _E cep 4748    Fr wfr 4795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-fr 4798
This theorem is referenced by:  tz7.7  5456  trelpss  36878
  Copyright terms: Public domain W3C validator