Theorem tospos 18375

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 18373	. 2 ⊢ (𝐹 ∈ Toset ↔ (𝐹 ∈ Poset ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑦 ∨ 𝑦(le‘𝐹)𝑥)))
4	3	simplbi 496	1 ⊢ (𝐹 ∈ Toset → 𝐹 ∈ Poset)

Syntax hints:   → wi 4   ∨ wo 848   ∈ wcel 2114  ∀wral 3052   class class class wbr 5086  ‘cfv 6492  Basecbs 17170  lecple 17218  Posetcpo 18264  Tosetctos 18371

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6448  df-fv 6500  df-toset 18372

This theorem is referenced by:  tltnle  18377  resstos  18387  omndadd2d  20096  omndadd2rd  20097  omndmul2  20099  omndmul  20101  gsumle  20111  orngsqr  20834  ofldchr  21566  odutos  33043  tlt3  33045  xrsclat  33086  isarchi3  33263  archirngz  33265  archiabllem1a  33267  archiabllem2c  33271  ordtrest2NEWlem  34082  ordtrest2NEW  34083  ordtconnlem1  34084  toslat  49469