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Theorem soirri 4769
Description: A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
Hypotheses
Ref Expression
soi.1  |-  R  Or  S
soi.2  |-  R  C_  ( S  X.  S
)
Assertion
Ref Expression
soirri  |-  -.  A R A

Proof of Theorem soirri
StepHypRef Expression
1 id 19 . 2  |-  ( A R A  ->  A R A )
2 soi.1 . . 3  |-  R  Or  S
3 soi.2 . . . . 5  |-  R  C_  ( S  X.  S
)
43brel 4438 . . . 4  |-  ( A R A  ->  ( A  e.  S  /\  A  e.  S )
)
54simpld 110 . . 3  |-  ( A R A  ->  A  e.  S )
6 sonr 4100 . . 3  |-  ( ( R  Or  S  /\  A  e.  S )  ->  -.  A R A )
72, 5, 6sylancr 405 . 2  |-  ( A R A  ->  -.  A R A )
81, 7pm2.65i 601 1  |-  -.  A R A
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1434    C_ wss 2982   class class class wbr 3805    Or wor 4078    X. cxp 4389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-po 4079  df-iso 4080  df-xp 4397
This theorem is referenced by:  son2lpi  4771  ltsonq  6702  genpdisj  6827  ltposr  7054  axpre-ltirr  7162
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