Theorem tospos 18342

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 2729	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2729	. . 3 ⊢ (le‘𝐹) = (le‘𝐹)
3	1, 2	istos 18340	. 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 2109  ∀wral 3044   class class class wbr 5095  ‘cfv 6486  Basecbs 17138  lecple 17186  Posetcpo 18231  Tosetctos 18338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5248

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-toset 18339

This theorem is referenced by:  tltnle  18344  resstos  18354  omndadd2d  20027  omndadd2rd  20028  omndmul2  20030  omndmul  20032  gsumle  20042  orngsqr  20769  ofldchr  21501  odutos  32923  tlt3  32925  xrsclat  32978  isarchi3  33139  archirngz  33141  archiabllem1a  33143  archiabllem2c  33147  ordtrest2NEWlem  33888  ordtrest2NEW  33889  ordtconnlem1  33890  toslat  48967