Theorem tospos 30671

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 2798	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2798	. . 3 ⊢ (le‘𝐹) = (le‘𝐹)
3	1, 2	istos 17637	. 2 ⊢ (𝐹 ∈ Toset ↔ (𝐹 ∈ Poset ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑦 ∨ 𝑦(le‘𝐹)𝑥)))
4	3	simplbi 501	1 ⊢ (𝐹 ∈ Toset → 𝐹 ∈ Poset)

Syntax hints:   → wi 4   ∨ wo 844   ∈ wcel 2111  ∀wral 3106   class class class wbr 5030  ‘cfv 6324  Basecbs 16475  lecple 16564  Posetcpo 17542  Tosetctos 17635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-toset 17636

This theorem is referenced by:  resstos  30673  tltnle  30675  odutos  30676  tlt3  30678  xrsclat  30714  omndadd2d  30759  omndadd2rd  30760  omndmul2  30763  omndmul  30765  gsumle  30775  isarchi3  30866  archirngz  30868  archiabllem1a  30870  archiabllem2c  30874  orngsqr  30928  ofldchr  30938  ordtrest2NEWlem  31275  ordtrest2NEW  31276  ordtconnlem1  31277