Theorem tospos 30645

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 2821	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2821	. . 3 ⊢ (le‘𝐹) = (le‘𝐹)
3	1, 2	istos 17645	. 2 ⊢ (𝐹 ∈ Toset ↔ (𝐹 ∈ Poset ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑦 ∨ 𝑦(le‘𝐹)𝑥)))
4	3	simplbi 500	1 ⊢ (𝐹 ∈ Toset → 𝐹 ∈ Poset)

Syntax hints:   → wi 4   ∨ wo 843   ∈ wcel 2114  ∀wral 3138   class class class wbr 5066  ‘cfv 6355  Basecbs 16483  lecple 16572  Posetcpo 17550  Tosetctos 17643

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-nul 5210

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-iota 6314  df-fv 6363  df-toset 17644

This theorem is referenced by:  resstos  30647  tltnle  30649  odutos  30650  tlt3  30652  xrsclat  30667  omndadd2d  30709  omndadd2rd  30710  omndmul2  30713  omndmul  30715  gsumle  30725  isarchi3  30816  archirngz  30818  archiabllem1a  30820  archiabllem2c  30824  orngsqr  30877  ofldchr  30887  ordtrest2NEWlem  31165  ordtrest2NEW  31166  ordtconnlem1  31167