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Theorem tospos 30649
 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 2824 . . 3 (Base‘𝐹) = (Base‘𝐹)
2 eqid 2824 . . 3 (le‘𝐹) = (le‘𝐹)
31, 2istos 17641 . 2 (𝐹 ∈ Toset ↔ (𝐹 ∈ Poset ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑦𝑦(le‘𝐹)𝑥)))
43simplbi 501 1 (𝐹 ∈ Toset → 𝐹 ∈ Poset)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 844   ∈ wcel 2115  ∀wral 3133   class class class wbr 5052  ‘cfv 6343  Basecbs 16479  lecple 16568  Posetcpo 17546  Tosetctos 17639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-toset 17640 This theorem is referenced by:  resstos  30651  tltnle  30653  odutos  30654  tlt3  30656  xrsclat  30692  omndadd2d  30734  omndadd2rd  30735  omndmul2  30738  omndmul  30740  gsumle  30750  isarchi3  30841  archirngz  30843  archiabllem1a  30845  archiabllem2c  30849  orngsqr  30902  ofldchr  30912  ordtrest2NEWlem  31190  ordtrest2NEW  31191  ordtconnlem1  31192
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