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Theorem tospos 27158
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 2460 . . 3  |-  ( Base `  F )  =  (
Base `  F )
2 eqid 2460 . . 3  |-  ( le
`  F )  =  ( le `  F
)
31, 2istos 15511 . 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 1762   A.wral 2807   class class class wbr 4440   ` cfv 5579   Basecbs 14479   lecple 14551   Posetcpo 15416  Tosetctos 15509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-nul 4569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-iota 5542  df-fv 5587  df-toset 15510
This theorem is referenced by:  resstos  27160  tltnle  27162  odutos  27163  tlt3  27165  xrsclat  27180  omndadd2d  27210  omndadd2rd  27211  omndmul2  27214  omndmul  27216  isarchi3  27243  archirngz  27245  archiabllem1a  27247  archiabllem2c  27251  gsumle  27283  orngsqr  27307  ofldchr  27317  ordtrest2NEWlem  27390  ordtrest2NEW  27391  ordtconlem1  27392
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