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

Theorem 0pos 17308
Description: Technical lemma to simplify the statement of ipopos 17514. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 16275) 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 5015 . 2 ∅ ∈ V
2 ral0 4299 . 2 𝑎 ∈ ∅ ∀𝑏 ∈ ∅ ∀𝑐 ∈ ∅ (𝑎𝑎 ∧ ((𝑎𝑏𝑏𝑎) → 𝑎 = 𝑏) ∧ ((𝑎𝑏𝑏𝑐) → 𝑎𝑐))
3 base0 16276 . . 3 ∅ = (Base‘∅)
4 df-ple 16326 . . . 4 le = Slot 10
54str0 16275 . . 3 ∅ = (le‘∅)
63, 5ispos 17301 . 2 (∅ ∈ Poset ↔ (∅ ∈ V ∧ ∀𝑎 ∈ ∅ ∀𝑏 ∈ ∅ ∀𝑐 ∈ ∅ (𝑎𝑎 ∧ ((𝑎𝑏𝑏𝑎) → 𝑎 = 𝑏) ∧ ((𝑎𝑏𝑏𝑐) → 𝑎𝑐))))
71, 2, 6mpbir2an 704 1 ∅ ∈ Poset
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113  wcel 2166  wral 3118  Vcvv 3415  c0 4145   class class class wbr 4874  0cc0 10253  1c1 10254  cdc 11822  lecple 16313  Posetcpo 17294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-iota 6087  df-fun 6126  df-fv 6132  df-slot 16227  df-base 16229  df-ple 16326  df-poset 17300
This theorem is referenced by:  ipopos  17514
  Copyright terms: Public domain W3C validator