MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  meet0 Structured version   Visualization version   GIF version

Theorem meet0 17742
Description: Lemma for odujoin 17747. (Contributed by Stefan O'Rear, 29-Jan-2015.) TODO (df-riota 7108 update): This proof increased from 152 bytes to 547 bytes after the df-riota 7108 change. Any way to shorten it? join0 17743 also.
Assertion
Ref Expression
meet0 (meet‘∅) = ∅

Proof of Theorem meet0
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 5208 . . 3 ∅ ∈ V
2 eqid 2826 . . . 4 (glb‘∅) = (glb‘∅)
3 eqid 2826 . . . 4 (meet‘∅) = (meet‘∅)
42, 3meetfval 17620 . . 3 (∅ ∈ V → (meet‘∅) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘∅)𝑧})
51, 4ax-mp 5 . 2 (meet‘∅) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘∅)𝑧}
6 df-oprab 7154 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘∅)𝑧} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)}
7 br0 5112 . . . . . . . . 9 ¬ {𝑥, 𝑦}∅𝑧
8 base0 16531 . . . . . . . . . . . . 13 ∅ = (Base‘∅)
9 eqid 2826 . . . . . . . . . . . . 13 (le‘∅) = (le‘∅)
10 biid 262 . . . . . . . . . . . . 13 ((∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦)) ↔ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦)))
11 id 22 . . . . . . . . . . . . 13 (∅ ∈ V → ∅ ∈ V)
128, 9, 2, 10, 11glbfval 17596 . . . . . . . . . . . 12 (∅ ∈ V → (glb‘∅) = ((𝑥 ∈ 𝒫 ∅ ↦ (𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦)))) ↾ {𝑥 ∣ ∃!𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦))}))
131, 12ax-mp 5 . . . . . . . . . . 11 (glb‘∅) = ((𝑥 ∈ 𝒫 ∅ ↦ (𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦)))) ↾ {𝑥 ∣ ∃!𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦))})
14 reu0 4322 . . . . . . . . . . . . . 14 ¬ ∃!𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦))
1514abf 4360 . . . . . . . . . . . . 13 {𝑥 ∣ ∃!𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦))} = ∅
1615reseq2i 5849 . . . . . . . . . . . 12 ((𝑥 ∈ 𝒫 ∅ ↦ (𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦)))) ↾ {𝑥 ∣ ∃!𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦))}) = ((𝑥 ∈ 𝒫 ∅ ↦ (𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦)))) ↾ ∅)
17 res0 5856 . . . . . . . . . . . 12 ((𝑥 ∈ 𝒫 ∅ ↦ (𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦)))) ↾ ∅) = ∅
1816, 17eqtri 2849 . . . . . . . . . . 11 ((𝑥 ∈ 𝒫 ∅ ↦ (𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦)))) ↾ {𝑥 ∣ ∃!𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦))}) = ∅
1913, 18eqtri 2849 . . . . . . . . . 10 (glb‘∅) = ∅
2019breqi 5069 . . . . . . . . 9 ({𝑥, 𝑦} (glb‘∅)𝑧 ↔ {𝑥, 𝑦}∅𝑧)
217, 20mtbir 324 . . . . . . . 8 ¬ {𝑥, 𝑦} (glb‘∅)𝑧
2221intnan 487 . . . . . . 7 ¬ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2322nex 1794 . . . . . 6 ¬ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2423nex 1794 . . . . 5 ¬ ∃𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2524nex 1794 . . . 4 ¬ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2625abf 4360 . . 3 {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)} = ∅
276, 26eqtri 2849 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘∅)𝑧} = ∅
285, 27eqtri 2849 1 (meet‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wex 1773  wcel 2107  {cab 2804  wral 3143  ∃!wreu 3145  Vcvv 3500  c0 4295  𝒫 cpw 4542  {cpr 4566  cop 4570   class class class wbr 5063  cmpt 5143  cres 5556  cfv 6354  crio 7107  {coprab 7151  lecple 16567  glbcglb 17548  meetcmee 17550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-oprab 7154  df-slot 16482  df-base 16484  df-glb 17580  df-meet 17582
This theorem is referenced by:  odumeet  17745
  Copyright terms: Public domain W3C validator