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Theorem 0pos 17162
Description: Technical lemma to simplify the statement of ipopos 17368. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 16118) 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 4925 . 2 ∅ ∈ V
2 ral0 4218 . 2 𝑎 ∈ ∅ ∀𝑏 ∈ ∅ ∀𝑐 ∈ ∅ (𝑎𝑎 ∧ ((𝑎𝑏𝑏𝑎) → 𝑎 = 𝑏) ∧ ((𝑎𝑏𝑏𝑐) → 𝑎𝑐))
3 base0 16119 . . 3 ∅ = (Base‘∅)
4 df-ple 16169 . . . 4 le = Slot 10
54str0 16118 . . 3 ∅ = (le‘∅)
63, 5ispos 17155 . 2 (∅ ∈ Poset ↔ (∅ ∈ V ∧ ∀𝑎 ∈ ∅ ∀𝑏 ∈ ∅ ∀𝑐 ∈ ∅ (𝑎𝑎 ∧ ((𝑎𝑏𝑏𝑎) → 𝑎 = 𝑏) ∧ ((𝑎𝑏𝑏𝑐) → 𝑎𝑐))))
71, 2, 6mpbir2an 690 1 ∅ ∈ Poset
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071  wcel 2145  wral 3061  Vcvv 3351  c0 4063   class class class wbr 4787  0cc0 10142  1c1 10143  cdc 11700  lecple 16156  Posetcpo 17148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-slot 16068  df-base 16070  df-ple 16169  df-poset 17154
This theorem is referenced by:  ipopos  17368
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