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Theorem soirri 5252
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 soi.1 . . . 4  |-  R  Or  S
2 sonr 4516 . . . 4  |-  ( ( R  Or  S  /\  A  e.  S )  ->  -.  A R A )
31, 2mpan 652 . . 3  |-  ( A  e.  S  ->  -.  A R A )
43adantl 453 . 2  |-  ( ( A  e.  S  /\  A  e.  S )  ->  -.  A R A )
5 soi.2 . . . 4  |-  R  C_  ( S  X.  S
)
65brel 4918 . . 3  |-  ( A R A  ->  ( A  e.  S  /\  A  e.  S )
)
76con3i 129 . 2  |-  ( -.  ( A  e.  S  /\  A  e.  S
)  ->  -.  A R A )
84, 7pm2.61i 158 1  |-  -.  A R A
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    e. wcel 1725    C_ wss 3312   class class class wbr 4204    Or wor 4494    X. cxp 4868
This theorem is referenced by:  son2lpi  5254  son2lpiOLD  5259  nqpr  8881  ltapr  8912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-po 4495  df-so 4496  df-xp 4876
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