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Theorem co02 4884
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 4869 . 2 Rel (𝐴 ∘ ∅)
2 rel0 4510 . 2 Rel ∅
3 noel 3271 . . . . . . 7 ¬ ⟨𝑥, 𝑧⟩ ∈ ∅
4 df-br 3806 . . . . . . 7 (𝑥𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ ∅)
53, 4mtbir 629 . . . . . 6 ¬ 𝑥𝑧
65intnanr 873 . . . . 5 ¬ (𝑥𝑧𝑧𝐴𝑦)
76nex 1430 . . . 4 ¬ ∃𝑧(𝑥𝑧𝑧𝐴𝑦)
8 vex 2613 . . . . 5 𝑥 ∈ V
9 vex 2613 . . . . 5 𝑦 ∈ V
108, 9opelco 4555 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥𝑧𝑧𝐴𝑦))
117, 10mtbir 629 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
12 noel 3271 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
1311, 122false 650 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
141, 2, 13eqrelriiv 4480 1 (𝐴 ∘ ∅) = ∅
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  wex 1422  wcel 1434  c0 3267  cop 3419   class class class wbr 3805  ccom 4395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-co 4400
This theorem is referenced by:  co01  4885
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