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Theorem co02 6280
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 6126 . 2 Rel (𝐴 ∘ ∅)
2 rel0 5809 . 2 Rel ∅
3 br0 5192 . . . . . 6 ¬ 𝑥𝑧
43intnanr 487 . . . . 5 ¬ (𝑥𝑧𝑧𝐴𝑦)
54nex 1800 . . . 4 ¬ ∃𝑧(𝑥𝑧𝑧𝐴𝑦)
6 vex 3484 . . . . 5 𝑥 ∈ V
7 vex 3484 . . . . 5 𝑦 ∈ V
86, 7opelco 5882 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥𝑧𝑧𝐴𝑦))
95, 8mtbir 323 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10 noel 4338 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
119, 102false 375 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
121, 2, 11eqrelriiv 5800 1 (𝐴 ∘ ∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wex 1779  wcel 2108  c0 4333  cop 4632   class class class wbr 5143  ccom 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-co 5694
This theorem is referenced by:  co01  6281  dfpo2  6316  relexpsucld  15073  gsumwmhm  18858  frmdgsum  18875  frmdup1  18877  efginvrel2  19745  0frgp  19797  evl1fval  22332  utop2nei  24259  tngds  24668  tngdsOLD  24669  tocycf  33137  tocyc01  33138  1arithidom  33565  mrsub0  35521  cononrel1  43607
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