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Theorem co02 5866
Description: Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
co02 (𝐴 ∘ ∅) = ∅

Proof of Theorem co02
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 5850 . 2 Rel (𝐴 ∘ ∅)
2 rel0 5445 . 2 Rel ∅
3 br0 4890 . . . . . 6 ¬ 𝑥𝑧
43intnanr 482 . . . . 5 ¬ (𝑥𝑧𝑧𝐴𝑦)
54nex 1896 . . . 4 ¬ ∃𝑧(𝑥𝑧𝑧𝐴𝑦)
6 vex 3386 . . . . 5 𝑥 ∈ V
7 vex 3386 . . . . 5 𝑦 ∈ V
86, 7opelco 5495 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥𝑧𝑧𝐴𝑦))
95, 8mtbir 315 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10 noel 4117 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
119, 102false 367 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
121, 2, 11eqrelriiv 5416 1 (𝐴 ∘ ∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 385   = wceq 1653  wex 1875  wcel 2157  c0 4113  cop 4372   class class class wbr 4841  ccom 5314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-xp 5316  df-rel 5317  df-co 5319
This theorem is referenced by:  co01  5867  gsumwmhm  17695  frmdgsum  17712  frmdup1  17714  efginvrel2  18450  0frgp  18504  evl1fval  20011  utop2nei  22379  tngds  22777  mrsub0  31922  dfpo2  32151  cononrel1  38671
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