Theorem 0pos 17566

Description: Technical lemma to simplify the statement of ipopos 17772. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 16537) 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.

Step	Hyp	Ref	Expression
1		0ex 5213	. 2 ⊢ ∅ ∈ V
2		ral0 4458	. 2 ⊢ ∀𝑎 ∈ ∅ ∀𝑏 ∈ ∅ ∀𝑐 ∈ ∅ (𝑎∅𝑎 ∧ ((𝑎∅𝑏 ∧ 𝑏∅𝑎) → 𝑎 = 𝑏) ∧ ((𝑎∅𝑏 ∧ 𝑏∅𝑐) → 𝑎∅𝑐))
3		base0 16538	. . 3 ⊢ ∅ = (Base‘∅)
4		df-ple 16587	. . . 4 ⊢ le = Slot ;10
5	4	str0 16537	. . 3 ⊢ ∅ = (le‘∅)
6	3, 5	ispos 17559	. 2 ⊢ (∅ ∈ Poset ↔ (∅ ∈ V ∧ ∀𝑎 ∈ ∅ ∀𝑏 ∈ ∅ ∀𝑐 ∈ ∅ (𝑎∅𝑎 ∧ ((𝑎∅𝑏 ∧ 𝑏∅𝑎) → 𝑎 = 𝑏) ∧ ((𝑎∅𝑏 ∧ 𝑏∅𝑐) → 𝑎∅𝑐))))
7	1, 2, 6	mpbir2an 709	1 ⊢ ∅ ∈ Poset

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2114  ∀wral 3140  Vcvv 3496  ∅c0 4293   class class class wbr 5068  0cc0 10539  1c1 10540  ;cdc 12101  lecple 16574  Posetcpo 17552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-slot 16489  df-base 16491  df-ple 16587  df-poset 17558

This theorem is referenced by:  ipopos  17772