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Theorem meet0 16960
 Description: Lemma for odujoin 16965. (Contributed by Stefan O'Rear, 29-Jan-2015.) TODO (df-riota 6511 update): This proof increased from 152 bytes to 547 bytes after the df-riota 6511 change. Any way to shorten it? join0 16961 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 4718 . . 3 ∅ ∈ V
2 eqid 2610 . . . 4 (glb‘∅) = (glb‘∅)
3 eqid 2610 . . . 4 (meet‘∅) = (meet‘∅)
42, 3meetfval 16838 . . 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 6553 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘∅)𝑧} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)}
7 br0 4631 . . . . . . . . 9 ¬ {𝑥, 𝑦}∅𝑧
8 base0 15740 . . . . . . . . . . . . 13 ∅ = (Base‘∅)
9 eqid 2610 . . . . . . . . . . . . 13 (le‘∅) = (le‘∅)
10 biid 250 . . . . . . . . . . . . 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 16814 . . . . . . . . . . . 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 3894 . . . . . . . . . . . . . . 15 ¬ ∃𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦))
15 reurex 3137 . . . . . . . . . . . . . . 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 187 . . . . . . . . . . . . . 14 ¬ ∃!𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦))
1716abf 3930 . . . . . . . . . . . . 13 {𝑥 ∣ ∃!𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦))} = ∅
1817reseq2i 5314 . . . . . . . . . . . 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 5321 . . . . . . . . . . . 12 ((𝑥 ∈ 𝒫 ∅ ↦ (𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦)))) ↾ ∅) = ∅
2018, 19eqtri 2632 . . . . . . . . . . 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 2632 . . . . . . . . . 10 (glb‘∅) = ∅
2221breqi 4589 . . . . . . . . 9 ({𝑥, 𝑦} (glb‘∅)𝑧 ↔ {𝑥, 𝑦}∅𝑧)
237, 22mtbir 312 . . . . . . . 8 ¬ {𝑥, 𝑦} (glb‘∅)𝑧
2423intnan 951 . . . . . . 7 ¬ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2524nex 1722 . . . . . 6 ¬ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2625nex 1722 . . . . 5 ¬ ∃𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2726nex 1722 . . . 4 ¬ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2827abf 3930 . . 3 {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)} = ∅
296, 28eqtri 2632 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘∅)𝑧} = ∅
305, 29eqtri 2632 1 (meet‘∅) = ∅
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  Vcvv 3173  ∅c0 3874  𝒫 cpw 4108  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   ↾ cres 5040  ‘cfv 5804  ℩crio 6510  {coprab 6550  lecple 15775  glbcglb 16766  meetcmee 16768 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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 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-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  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-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-oprab 6553  df-slot 15699  df-base 15700  df-glb 16798  df-meet 16800 This theorem is referenced by:  odumeet  16963
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