Theorem tospos 18624

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 2763	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2763	. . 3 ⊢ (le‘𝐹) = (le‘𝐹)
3	1, 2	istos 18622	. 2 ⊢ (𝐹 ∈ Toset ↔ (𝐹 ∈ Poset ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑦 ∨ 𝑦(le‘𝐹)𝑥)))
4	3	simplbi 502	1 ⊢ (𝐹 ∈ Toset → 𝐹 ∈ Poset)

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   ∈ wcel 2147  ∀wral 3080   class class class wbr 5106  ‘cfv 6561  Basecbs 17419  lecple 17467  Posetcpo 18513  Tosetctos 18620

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5264

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fv 6569  df-toset 18621

This theorem is used by:  tltnle  18626  resstos  18636  omndadd2d  20377  omndadd2rd  20378  omndmul2  20380  omndmul  20382  gsumle  20392  orngsqr  21156  ofldchr  21915  odutos  33602  tlt3  33604  xrsclat  33648  isarchi3  33824  archirngz  33826  archiabllem1a  33828  archiabllem2c  33832  ordtrest2NEWlem  34637  ordtrest2NEW  34638  ordtconnlem1  34639  toslat  50311