Theorem tospos 18321

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 2731	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2731	. . 3 ⊢ (le‘𝐹) = (le‘𝐹)
3	1, 2	istos 18319	. 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 2111  ∀wral 3047   class class class wbr 5091  ‘cfv 6481  Basecbs 17117  lecple 17165  Posetcpo 18210  Tosetctos 18317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5244

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-iota 6437  df-fv 6489  df-toset 18318

This theorem is referenced by:  tltnle  18323  resstos  18333  omndadd2d  20040  omndadd2rd  20041  omndmul2  20043  omndmul  20045  gsumle  20055  orngsqr  20779  ofldchr  21511  odutos  32944  tlt3  32946  xrsclat  32987  isarchi3  33151  archirngz  33153  archiabllem1a  33155  archiabllem2c  33159  ordtrest2NEWlem  33930  ordtrest2NEW  33931  ordtconnlem1  33932  toslat  49012