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Theorem 0pos 16777
 Description: Technical lemma to simplify the statement of ipopos 16983. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 15739) 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 4718 . 2 ∅ ∈ V
2 ral0 4028 . 2 𝑎 ∈ ∅ ∀𝑏 ∈ ∅ ∀𝑐 ∈ ∅ (𝑎𝑎 ∧ ((𝑎𝑏𝑏𝑎) → 𝑎 = 𝑏) ∧ ((𝑎𝑏𝑏𝑐) → 𝑎𝑐))
3 base0 15740 . . 3 ∅ = (Base‘∅)
4 df-ple 15788 . . . 4 le = Slot 10
54str0 15739 . . 3 ∅ = (le‘∅)
63, 5ispos 16770 . 2 (∅ ∈ Poset ↔ (∅ ∈ V ∧ ∀𝑎 ∈ ∅ ∀𝑏 ∈ ∅ ∀𝑐 ∈ ∅ (𝑎𝑎 ∧ ((𝑎𝑏𝑏𝑎) → 𝑎 = 𝑏) ∧ ((𝑎𝑏𝑏𝑐) → 𝑎𝑐))))
71, 2, 6mpbir2an 957 1 ∅ ∈ Poset
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ∅c0 3874   class class class wbr 4583  0cc0 9815  1c1 9816  ;cdc 11369  lecple 15775  Posetcpo 16763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-slot 15699  df-base 15700  df-ple 15788  df-poset 16769 This theorem is referenced by:  ipopos  16983
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