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Theorem tospos 28098
Description: A Toset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Assertion
Ref Expression
tospos  |-  ( F  e. Toset  ->  F  e.  Poset )

Proof of Theorem tospos
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2402 . . 3  |-  ( Base `  F )  =  (
Base `  F )
2 eqid 2402 . . 3  |-  ( le
`  F )  =  ( le `  F
)
31, 2istos 15989 . 2  |-  ( F  e. Toset 
<->  ( F  e.  Poset  /\ 
A. x  e.  (
Base `  F ) A. y  e.  ( Base `  F ) ( x ( le `  F ) y  \/  y ( le `  F ) x ) ) )
43simplbi 458 1  |-  ( F  e. Toset  ->  F  e.  Poset )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    e. wcel 1842   A.wral 2754   class class class wbr 4395   ` cfv 5569   Basecbs 14841   lecple 14916   Posetcpo 15893  Tosetctos 15987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4525
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-toset 15988
This theorem is referenced by:  resstos  28100  tltnle  28102  odutos  28103  tlt3  28105  xrsclat  28120  omndadd2d  28150  omndadd2rd  28151  omndmul2  28154  omndmul  28156  isarchi3  28183  archirngz  28185  archiabllem1a  28187  archiabllem2c  28191  gsumle  28221  orngsqr  28247  ofldchr  28257  ordtrest2NEWlem  28357  ordtrest2NEW  28358  ordtconlem1  28359
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