Theorem tospos 18462

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 2765	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2765	. . 3 ⊢ (le‘𝐹) = (le‘𝐹)
3	1, 2	istos 18460	. 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 860   ∈ wcel 2145  ∀wral 3079   class class class wbr 5104  ‘cfv 6525  Basecbs 17257  lecple 17305  Posetcpo 18351  Tosetctos 18458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5260

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-iota 6481  df-fv 6533  df-toset 18459

This theorem is referenced by:  tltnle  18464  resstos  18474  omndadd2d  20188  omndadd2rd  20189  omndmul2  20191  omndmul  20193  gsumle  20203  orngsqr  20935  ofldchr  21683  odutos  33196  tlt3  33198  xrsclat  33239  isarchi3  33415  archirngz  33417  archiabllem1a  33419  archiabllem2c  33423  ordtrest2NEWlem  34224  ordtrest2NEW  34225  ordtconnlem1  34226  toslat  49612