Theorem tospos 18382

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 2740	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2740	. . 3 ⊢ (le‘𝐹) = (le‘𝐹)
3	1, 2	istos 18380	. 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 853   ∈ wcel 2119  ∀wral 3054   class class class wbr 5079  ‘cfv 6492  Basecbs 17177  lecple 17225  Posetcpo 18271  Tosetctos 18378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-toset 18379

This theorem is referenced by:  tltnle  18384  resstos  18394  omndadd2d  20103  omndadd2rd  20104  omndmul2  20106  omndmul  20108  gsumle  20118  orngsqr  20845  ofldchr  21558  odutos  33054  tlt3  33056  xrsclat  33097  isarchi3  33275  archirngz  33277  archiabllem1a  33279  archiabllem2c  33283  ordtrest2NEWlem  34113  ordtrest2NEW  34114  ordtconnlem1  34115  toslat  49479