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Theorem co02 5124
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 5109 . 2 Rel (𝐴 ∘ ∅)
2 rel0 4736 . 2 Rel ∅
3 noel 3418 . . . . . . 7 ¬ ⟨𝑥, 𝑧⟩ ∈ ∅
4 df-br 3990 . . . . . . 7 (𝑥𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ ∅)
53, 4mtbir 666 . . . . . 6 ¬ 𝑥𝑧
65intnanr 925 . . . . 5 ¬ (𝑥𝑧𝑧𝐴𝑦)
76nex 1493 . . . 4 ¬ ∃𝑧(𝑥𝑧𝑧𝐴𝑦)
8 vex 2733 . . . . 5 𝑥 ∈ V
9 vex 2733 . . . . 5 𝑦 ∈ V
108, 9opelco 4783 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥𝑧𝑧𝐴𝑦))
117, 10mtbir 666 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
12 noel 3418 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
1311, 122false 696 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
141, 2, 13eqrelriiv 4705 1 (𝐴 ∘ ∅) = ∅
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1348  wex 1485  wcel 2141  c0 3414  cop 3586   class class class wbr 3989  ccom 4615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-co 4620
This theorem is referenced by:  co01  5125
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