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Theorem co02 4859
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 4844 . 2 Rel (𝐴 ∘ ∅)
2 rel0 4487 . 2 Rel ∅
3 noel 3253 . . . . . . 7 ¬ ⟨𝑥, 𝑧⟩ ∈ ∅
4 df-br 3790 . . . . . . 7 (𝑥𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ ∅)
53, 4mtbir 604 . . . . . 6 ¬ 𝑥𝑧
65intnanr 848 . . . . 5 ¬ (𝑥𝑧𝑧𝐴𝑦)
76nex 1403 . . . 4 ¬ ∃𝑧(𝑥𝑧𝑧𝐴𝑦)
8 vex 2575 . . . . 5 𝑥 ∈ V
9 vex 2575 . . . . 5 𝑦 ∈ V
108, 9opelco 4532 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥𝑧𝑧𝐴𝑦))
117, 10mtbir 604 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
12 noel 3253 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
1311, 122false 625 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
141, 2, 13eqrelriiv 4459 1 (𝐴 ∘ ∅) = ∅
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1257  wex 1395  wcel 1407  c0 3249  cop 3403   class class class wbr 3789  ccom 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3790  df-opab 3844  df-xp 4376  df-rel 4377  df-co 4379
This theorem is referenced by:  co01  4860
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