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

Theorem meet0 17184
Description: Lemma for odujoin 17189. (Contributed by Stefan O'Rear, 29-Jan-2015.) TODO (df-riota 6651 update): This proof increased from 152 bytes to 547 bytes after the df-riota 6651 change. Any way to shorten it? join0 17185 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 4823 . . 3 ∅ ∈ V
2 eqid 2651 . . . 4 (glb‘∅) = (glb‘∅)
3 eqid 2651 . . . 4 (meet‘∅) = (meet‘∅)
42, 3meetfval 17062 . . 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 6694 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘∅)𝑧} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)}
7 br0 4734 . . . . . . . . 9 ¬ {𝑥, 𝑦}∅𝑧
8 base0 15959 . . . . . . . . . . . . 13 ∅ = (Base‘∅)
9 eqid 2651 . . . . . . . . . . . . 13 (le‘∅) = (le‘∅)
10 biid 251 . . . . . . . . . . . . 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 17038 . . . . . . . . . . . 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 rex0 3971 . . . . . . . . . . . . . . 15 ¬ ∃𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦))
15 reurex 3190 . . . . . . . . . . . . . . 15 (∃!𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦)) → ∃𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦)))
1614, 15mto 188 . . . . . . . . . . . . . 14 ¬ ∃!𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦))
1716abf 4011 . . . . . . . . . . . . 13 {𝑥 ∣ ∃!𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦))} = ∅
1817reseq2i 5425 . . . . . . . . . . . 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)))) ↾ ∅)
19 res0 5432 . . . . . . . . . . . 12 ((𝑥 ∈ 𝒫 ∅ ↦ (𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦)))) ↾ ∅) = ∅
2018, 19eqtri 2673 . . . . . . . . . . 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))}) = ∅
2113, 20eqtri 2673 . . . . . . . . . 10 (glb‘∅) = ∅
2221breqi 4691 . . . . . . . . 9 ({𝑥, 𝑦} (glb‘∅)𝑧 ↔ {𝑥, 𝑦}∅𝑧)
237, 22mtbir 312 . . . . . . . 8 ¬ {𝑥, 𝑦} (glb‘∅)𝑧
2423intnan 980 . . . . . . 7 ¬ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2524nex 1771 . . . . . 6 ¬ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2625nex 1771 . . . . 5 ¬ ∃𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2726nex 1771 . . . 4 ¬ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2827abf 4011 . . 3 {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)} = ∅
296, 28eqtri 2673 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘∅)𝑧} = ∅
305, 29eqtri 2673 1 (meet‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wral 2941  wrex 2942  ∃!wreu 2943  Vcvv 3231  c0 3948  𝒫 cpw 4191  {cpr 4212  cop 4216   class class class wbr 4685  cmpt 4762  cres 5145  cfv 5926  crio 6650  {coprab 6691  lecple 15995  glbcglb 16990  meetcmee 16992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-oprab 6694  df-slot 15908  df-base 15910  df-glb 17022  df-meet 17024
This theorem is referenced by:  odumeet  17187
  Copyright terms: Public domain W3C validator