Theorem tospos 18419

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 2728	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2728	. . 3 ⊢ (le‘𝐹) = (le‘𝐹)
3	1, 2	istos 18417	. 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 845   ∈ wcel 2098  ∀wral 3058   class class class wbr 5152  ‘cfv 6553  Basecbs 17187  lecple 17247  Posetcpo 18306  Tosetctos 18415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-nul 5310

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-toset 18416

This theorem is referenced by:  tltnle  18421  resstos  32715  odutos  32716  tlt3  32718  xrsclat  32759  omndadd2d  32809  omndadd2rd  32810  omndmul2  32813  omndmul  32815  gsumle  32825  isarchi3  32916  archirngz  32918  archiabllem1a  32920  archiabllem2c  32924  orngsqr  33043  ofldchr  33053  ordtrest2NEWlem  33556  ordtrest2NEW  33557  ordtconnlem1  33558  toslat  48071