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Theorem tospos 28795
Description: A Toset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Assertion
Ref Expression
tospos (𝐹 ∈ Toset → 𝐹 ∈ Poset)

Proof of Theorem tospos
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2609 . . 3 (Base‘𝐹) = (Base‘𝐹)
2 eqid 2609 . . 3 (le‘𝐹) = (le‘𝐹)
31, 2istos 16804 . 2 (𝐹 ∈ Toset ↔ (𝐹 ∈ Poset ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥)))
43simplbi 474 1 (𝐹 ∈ Toset → 𝐹 ∈ Poset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 381  wcel 1976  wral 2895   class class class wbr 4577  cfv 5790  Basecbs 15641  lecple 15721  Posetcpo 16709  Tosetctos 16802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-toset 16803
This theorem is referenced by:  resstos  28797  tltnle  28799  odutos  28800  tlt3  28802  xrsclat  28817  omndadd2d  28845  omndadd2rd  28846  omndmul2  28849  omndmul  28851  isarchi3  28878  archirngz  28880  archiabllem1a  28882  archiabllem2c  28886  gsumle  28916  orngsqr  28941  ofldchr  28951  ordtrest2NEWlem  29102  ordtrest2NEW  29103  ordtconlem1  29104
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