MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  soirriOLD Unicode version

Theorem soirriOLD 5090
Description: A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
soiOLD.1  |-  A  e. 
_V
soiOLD.2  |-  R  Or  S
soiOLD.3  |-  R  C_  ( S  X.  S
)
Assertion
Ref Expression
soirriOLD  |-  -.  A R A

Proof of Theorem soirriOLD
StepHypRef Expression
1 soiOLD.2 . . . 4  |-  R  Or  S
2 sonr 4351 . . . 4  |-  ( ( R  Or  S  /\  A  e.  S )  ->  -.  A R A )
31, 2mpan 651 . . 3  |-  ( A  e.  S  ->  -.  A R A )
43adantl 452 . 2  |-  ( ( A  e.  S  /\  A  e.  S )  ->  -.  A R A )
5 soiOLD.3 . . . 4  |-  R  C_  ( S  X.  S
)
65brel 4753 . . 3  |-  ( A R A  ->  ( A  e.  S  /\  A  e.  S )
)
76con3i 127 . 2  |-  ( -.  ( A  e.  S  /\  A  e.  S
)  ->  -.  A R A )
84, 7pm2.61i 156 1  |-  -.  A R A
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    e. wcel 1696   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    Or wor 4329    X. cxp 4703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-po 4330  df-so 4331  df-xp 4711
  Copyright terms: Public domain W3C validator