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Theorem co02 5160
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 5145 . 2 Rel (𝐴 ∘ ∅)
2 rel0 4769 . 2 Rel ∅
3 noel 3441 . . . . . . 7 ¬ ⟨𝑥, 𝑧⟩ ∈ ∅
4 df-br 4019 . . . . . . 7 (𝑥𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ ∅)
53, 4mtbir 672 . . . . . 6 ¬ 𝑥𝑧
65intnanr 931 . . . . 5 ¬ (𝑥𝑧𝑧𝐴𝑦)
76nex 1511 . . . 4 ¬ ∃𝑧(𝑥𝑧𝑧𝐴𝑦)
8 vex 2755 . . . . 5 𝑥 ∈ V
9 vex 2755 . . . . 5 𝑦 ∈ V
108, 9opelco 4817 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥𝑧𝑧𝐴𝑦))
117, 10mtbir 672 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
12 noel 3441 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
1311, 122false 702 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
141, 2, 13eqrelriiv 4738 1 (𝐴 ∘ ∅) = ∅
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1364  wex 1503  wcel 2160  c0 3437  cop 3610   class class class wbr 4018  ccom 4648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-co 4653
This theorem is referenced by:  co01  5161
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