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Theorem meet0 17863
Description: Lemma for odujoin 17868. (Contributed by Stefan O'Rear, 29-Jan-2015.) TODO (df-riota 7127 update): This proof increased from 152 bytes to 547 bytes after the df-riota 7127 change. Any way to shorten it? join0 17864 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 5175 . . 3 ∅ ∈ V
2 eqid 2738 . . . 4 (glb‘∅) = (glb‘∅)
3 eqid 2738 . . . 4 (meet‘∅) = (meet‘∅)
42, 3meetfval 17741 . . 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 7174 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘∅)𝑧} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)}
7 br0 5079 . . . . . . . . 9 ¬ {𝑥, 𝑦}∅𝑧
8 base0 16639 . . . . . . . . . . . . 13 ∅ = (Base‘∅)
9 eqid 2738 . . . . . . . . . . . . 13 (le‘∅) = (le‘∅)
10 biid 264 . . . . . . . . . . . . 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 17717 . . . . . . . . . . . 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 4247 . . . . . . . . . . . . . 14 ¬ ∃!𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦))
1514abf 4291 . . . . . . . . . . . . 13 {𝑥 ∣ ∃!𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦))} = ∅
1615reseq2i 5822 . . . . . . . . . . . 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 5829 . . . . . . . . . . . 12 ((𝑥 ∈ 𝒫 ∅ ↦ (𝑦 ∈ ∅ (∀𝑧𝑥 𝑦(le‘∅)𝑧 ∧ ∀𝑤 ∈ ∅ (∀𝑧𝑥 𝑤(le‘∅)𝑧𝑤(le‘∅)𝑦)))) ↾ ∅) = ∅
1816, 17eqtri 2761 . . . . . . . . . . 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 2761 . . . . . . . . . 10 (glb‘∅) = ∅
2019breqi 5036 . . . . . . . . 9 ({𝑥, 𝑦} (glb‘∅)𝑧 ↔ {𝑥, 𝑦}∅𝑧)
217, 20mtbir 326 . . . . . . . 8 ¬ {𝑥, 𝑦} (glb‘∅)𝑧
2221intnan 490 . . . . . . 7 ¬ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2322nex 1807 . . . . . 6 ¬ ∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2423nex 1807 . . . . 5 ¬ ∃𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2524nex 1807 . . . 4 ¬ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)
2625abf 4291 . . 3 {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ {𝑥, 𝑦} (glb‘∅)𝑧)} = ∅
276, 26eqtri 2761 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ {𝑥, 𝑦} (glb‘∅)𝑧} = ∅
285, 27eqtri 2761 1 (meet‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wex 1786  wcel 2114  {cab 2716  wral 3053  ∃!wreu 3055  Vcvv 3398  c0 4211  𝒫 cpw 4488  {cpr 4518  cop 4522   class class class wbr 5030  cmpt 5110  cres 5527  cfv 6339  crio 7126  {coprab 7171  lecple 16675  glbcglb 17669  meetcmee 17671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-oprab 7174  df-slot 16590  df-base 16592  df-glb 17701  df-meet 17703
This theorem is referenced by:  odumeet  17866
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