Theorem tospos 18386

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 2730	. . 3 ⊢ (Base‘𝐹) = (Base‘𝐹)
2		eqid 2730	. . 3 ⊢ (le‘𝐹) = (le‘𝐹)
3	1, 2	istos 18384	. 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 847   ∈ wcel 2109  ∀wral 3045   class class class wbr 5110  ‘cfv 6514  Basecbs 17186  lecple 17234  Posetcpo 18275  Tosetctos 18382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-toset 18383

This theorem is referenced by:  tltnle  18388  resstos  32900  odutos  32901  tlt3  32903  xrsclat  32956  omndadd2d  33029  omndadd2rd  33030  omndmul2  33033  omndmul  33035  gsumle  33045  isarchi3  33148  archirngz  33150  archiabllem1a  33152  archiabllem2c  33156  orngsqr  33289  ofldchr  33299  ordtrest2NEWlem  33919  ordtrest2NEW  33920  ordtconnlem1  33921  toslat  48974