MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0pos Structured version   Visualization version   GIF version

Theorem 0pos 17001
Description: Technical lemma to simplify the statement of ipopos 17207. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 15958) and any relation partially orders an empty set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
0pos ∅ ∈ Poset

Proof of Theorem 0pos
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4823 . 2 ∅ ∈ V
2 ral0 4109 . 2 𝑎 ∈ ∅ ∀𝑏 ∈ ∅ ∀𝑐 ∈ ∅ (𝑎𝑎 ∧ ((𝑎𝑏𝑏𝑎) → 𝑎 = 𝑏) ∧ ((𝑎𝑏𝑏𝑐) → 𝑎𝑐))
3 base0 15959 . . 3 ∅ = (Base‘∅)
4 df-ple 16008 . . . 4 le = Slot 10
54str0 15958 . . 3 ∅ = (le‘∅)
63, 5ispos 16994 . 2 (∅ ∈ Poset ↔ (∅ ∈ V ∧ ∀𝑎 ∈ ∅ ∀𝑏 ∈ ∅ ∀𝑐 ∈ ∅ (𝑎𝑎 ∧ ((𝑎𝑏𝑏𝑎) → 𝑎 = 𝑏) ∧ ((𝑎𝑏𝑏𝑐) → 𝑎𝑐))))
71, 2, 6mpbir2an 975 1 ∅ ∈ Poset
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054  wcel 2030  wral 2941  Vcvv 3231  c0 3948   class class class wbr 4685  0cc0 9974  1c1 9975  cdc 11531  lecple 15995  Posetcpo 16987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-slot 15908  df-base 15910  df-ple 16008  df-poset 16993
This theorem is referenced by:  ipopos  17207
  Copyright terms: Public domain W3C validator