Theorem tospos 18433

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 2761	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2761	. . 3 ⊢ (le‘𝐹) = (le‘𝐹)
3	1, 2	istos 18431	. 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 858   ∈ wcel 2141  ∀wral 3075   class class class wbr 5099  ‘cfv 6517  Basecbs 17228  lecple 17276  Posetcpo 18322  Tosetctos 18429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5255

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6473  df-fv 6525  df-toset 18430

This theorem is referenced by:  tltnle  18435  resstos  18445  omndadd2d  20153  omndadd2rd  20154  omndmul2  20156  omndmul  20158  gsumle  20168  orngsqr  20895  ofldchr  21608  odutos  33107  tlt3  33109  xrsclat  33150  isarchi3  33328  archirngz  33330  archiabllem1a  33332  archiabllem2c  33336  ordtrest2NEWlem  34180  ordtrest2NEW  34181  ordtconnlem1  34182  toslat  49567