Theorem tospos 18430

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 18428	. 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 2108  ∀wral 3051   class class class wbr 5119  ‘cfv 6531  Basecbs 17228  lecple 17278  Posetcpo 18319  Tosetctos 18426

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539  df-toset 18427

This theorem is referenced by:  tltnle  18432  resstos  32947  odutos  32948  tlt3  32950  xrsclat  33003  omndadd2d  33076  omndadd2rd  33077  omndmul2  33080  omndmul  33082  gsumle  33092  isarchi3  33185  archirngz  33187  archiabllem1a  33189  archiabllem2c  33193  orngsqr  33326  ofldchr  33336  ordtrest2NEWlem  33953  ordtrest2NEW  33954  ordtconnlem1  33955  toslat  48956