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

Theorem soirri 5335
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 4773 . . . 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 4998 . . 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 1758    C_ wss 3439   class class class wbr 4403    Or wor 4751    X. cxp 4949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-po 4752  df-so 4753  df-xp 4957
This theorem is referenced by:  son2lpi  5337  son2lpiOLD  5342  nqpr  9297  ltapr  9328
  Copyright terms: Public domain W3C validator