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Theorem tospos 26253
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 2451 . . 3  |-  ( Base `  F )  =  (
Base `  F )
2 eqid 2451 . . 3  |-  ( le
`  F )  =  ( le `  F
)
31, 2istos 15307 . 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 460 1  |-  ( F  e. Toset  ->  F  e.  Poset )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    e. wcel 1758   A.wral 2795   class class class wbr 4390   ` cfv 5516   Basecbs 14276   lecple 14347   Posetcpo 15212  Tosetctos 15305
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4519
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-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-iota 5479  df-fv 5524  df-toset 15306
This theorem is referenced by:  resstos  26255  tltnle  26257  odutos  26258  tlt3  26260  xrsclat  26275  omndadd2d  26305  omndadd2rd  26306  omndmul2  26309  omndmul  26311  isarchi3  26338  archirngz  26340  archiabllem1a  26342  archiabllem2c  26346  gsumle  26380  orngsqr  26406  ofldchr  26416  ordtrest2NEWlem  26486  ordtrest2NEW  26487  ordtconlem1  26488
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