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Theorem soirri 5018
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 4674 . . . 4  |-  ( A R A  ->  ( A  e.  S  /\  A  e.  S )
)
54simpld 112 . . 3  |-  ( A R A  ->  A  e.  S )
6 sonr 4313 . . 3  |-  ( ( R  Or  S  /\  A  e.  S )  ->  -.  A R A )
72, 5, 6sylancr 414 . 2  |-  ( A R A  ->  -.  A R A )
81, 7pm2.65i 639 1  |-  -.  A R A
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 2148    C_ wss 3129   class class class wbr 4000    Or wor 4291    X. cxp 4620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-po 4292  df-iso 4293  df-xp 4628
This theorem is referenced by:  son2lpi  5020  ltsonq  7375  genpdisj  7500  ltposr  7740  axpre-ltirr  7859
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