Theorem tospos 30647

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 2823	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2823	. . 3 ⊢ (le‘𝐹) = (le‘𝐹)
3	1, 2	istos 17647	. 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 3140   class class class wbr 5068  ‘cfv 6357  Basecbs 16485  lecple 16574  Posetcpo 17552  Tosetctos 17645

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 2795  ax-nul 5212

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 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-toset 17646

This theorem is referenced by:  resstos  30649  tltnle  30651  odutos  30652  tlt3  30654  xrsclat  30669  omndadd2d  30711  omndadd2rd  30712  omndmul2  30715  omndmul  30717  gsumle  30727  isarchi3  30818  archirngz  30820  archiabllem1a  30822  archiabllem2c  30826  orngsqr  30879  ofldchr  30889  ordtrest2NEWlem  31167  ordtrest2NEW  31168  ordtconnlem1  31169