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Theorem co02 5566
 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 5550 . 2 Rel (𝐴 ∘ ∅)
2 rel0 5166 . 2 Rel ∅
3 br0 4631 . . . . . 6 ¬ 𝑥𝑧
43intnanr 952 . . . . 5 ¬ (𝑥𝑧𝑧𝐴𝑦)
54nex 1722 . . . 4 ¬ ∃𝑧(𝑥𝑧𝑧𝐴𝑦)
6 vex 3176 . . . . 5 𝑥 ∈ V
7 vex 3176 . . . . 5 𝑦 ∈ V
86, 7opelco 5215 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥𝑧𝑧𝐴𝑦))
95, 8mtbir 312 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10 noel 3878 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
119, 102false 364 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
121, 2, 11eqrelriiv 5137 1 (𝐴 ∘ ∅) = ∅
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∅c0 3874  ⟨cop 4131   class class class wbr 4583   ∘ ccom 5042 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 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-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-co 5047 This theorem is referenced by:  co01  5567  gsumwmhm  17205  frmdgsum  17222  frmdup1  17224  efginvrel2  17963  0frgp  18015  evl1fval  19513  utop2nei  21864  tngds  22262  mrsub0  30667  dfpo2  30898  cononrel1  36919
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