Theorem tospos 18478

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.

Step	Hyp	Ref	Expression
1		eqid 2735	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2735	. . 3 ⊢ (le‘𝐹) = (le‘𝐹)
3	1, 2	istos 18476	. 2 ⊢ (𝐹 ∈ Toset ↔ (𝐹 ∈ Poset ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑦 ∨ 𝑦(le‘𝐹)𝑥)))
4	3	simplbi 497	1 ⊢ (𝐹 ∈ Toset → 𝐹 ∈ Poset)

Syntax hints:   → wi 4   ∨ wo 847   ∈ wcel 2106  ∀wral 3059   class class class wbr 5148  ‘cfv 6563  Basecbs 17245  lecple 17305  Posetcpo 18365  Tosetctos 18474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-toset 18475

This theorem is referenced by:  tltnle  18480  resstos  32942  odutos  32943  tlt3  32945  xrsclat  32996  omndadd2d  33068  omndadd2rd  33069  omndmul2  33072  omndmul  33074  gsumle  33084  isarchi3  33177  archirngz  33179  archiabllem1a  33181  archiabllem2c  33185  orngsqr  33314  ofldchr  33324  ordtrest2NEWlem  33883  ordtrest2NEW  33884  ordtconnlem1  33885  toslat  48771