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Theorem soirri 5383
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 4811 . . . 4  |-  ( ( R  Or  S  /\  A  e.  S )  ->  -.  A R A )
31, 2mpan 670 . . 3  |-  ( A  e.  S  ->  -.  A R A )
43adantl 466 . 2  |-  ( ( A  e.  S  /\  A  e.  S )  ->  -.  A R A )
5 soi.2 . . . 4  |-  R  C_  ( S  X.  S
)
65brel 5038 . . 3  |-  ( A R A  ->  ( A  e.  S  /\  A  e.  S )
)
76con3i 135 . 2  |-  ( -.  ( A  e.  S  /\  A  e.  S
)  ->  -.  A R A )
84, 7pm2.61i 164 1  |-  -.  A R A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    e. wcel 1804    C_ wss 3461   class class class wbr 4437    Or wor 4789    X. cxp 4987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-po 4790  df-so 4791  df-xp 4995
This theorem is referenced by:  son2lpi  5385  son2lpiOLD  5390  nqpr  9395  ltapr  9426
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