Theorem tospos 18326

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 2733	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2733	. . 3 ⊢ (le‘𝐹) = (le‘𝐹)
3	1, 2	istos 18324	. 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 2113  ∀wral 3048   class class class wbr 5093  ‘cfv 6486  Basecbs 17122  lecple 17170  Posetcpo 18215  Tosetctos 18322

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-nul 5246

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6442  df-fv 6494  df-toset 18323

This theorem is referenced by:  tltnle  18328  resstos  18338  omndadd2d  20044  omndadd2rd  20045  omndmul2  20047  omndmul  20049  gsumle  20059  orngsqr  20783  ofldchr  21515  odutos  32956  tlt3  32958  xrsclat  32999  isarchi3  33163  archirngz  33165  archiabllem1a  33167  archiabllem2c  33171  ordtrest2NEWlem  33956  ordtrest2NEW  33957  ordtconnlem1  33958  toslat  49106