Theorem tospos 18475

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 2763	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2763	. . 3 ⊢ (le‘𝐹) = (le‘𝐹)
3	1, 2	istos 18473	. 2 ⊢ (𝐹 ∈ Toset ↔ (𝐹 ∈ Poset ∧ ∀𝑥 ∈ (Base‘𝐹)∀𝑦 ∈ (Base‘𝐹)(𝑥(le‘𝐹)𝑦 ∨ 𝑦(le‘𝐹)𝑥)))
4	3	simplbi 501	1 ⊢ (𝐹 ∈ Toset → 𝐹 ∈ Poset)

Syntax hints:   → wi 4   ∨ wo 860   ∈ wcel 2143  ∀wral 3079   class class class wbr 5110  ‘cfv 6538  Basecbs 17270  lecple 17318  Posetcpo 18364  Tosetctos 18471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-toset 18472

This theorem is referenced by:  tltnle  18477  resstos  18487  omndadd2d  20201  omndadd2rd  20202  omndmul2  20204  omndmul  20206  gsumle  20216  orngsqr  20950  ofldchr  21707  odutos  33269  tlt3  33271  xrsclat  33312  isarchi3  33488  archirngz  33490  archiabllem1a  33492  archiabllem2c  33496  ordtrest2NEWlem  34293  ordtrest2NEW  34294  ordtconnlem1  34295  toslat  49743