Theorem tospos 18465

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 2737	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2737	. . 3 ⊢ (le‘𝐹) = (le‘𝐹)
3	1, 2	istos 18463	. 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 848   ∈ wcel 2108  ∀wral 3061   class class class wbr 5143  ‘cfv 6561  Basecbs 17247  lecple 17304  Posetcpo 18353  Tosetctos 18461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-toset 18462

This theorem is referenced by:  tltnle  18467  resstos  32957  odutos  32958  tlt3  32960  xrsclat  33013  omndadd2d  33085  omndadd2rd  33086  omndmul2  33089  omndmul  33091  gsumle  33101  isarchi3  33194  archirngz  33196  archiabllem1a  33198  archiabllem2c  33202  orngsqr  33334  ofldchr  33344  ordtrest2NEWlem  33921  ordtrest2NEW  33922  ordtconnlem1  33923  toslat  48871