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Theorem co02 5020
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 5005 . 2 Rel (𝐴 ∘ ∅)
2 rel0 4632 . 2 Rel ∅
3 noel 3335 . . . . . . 7 ¬ ⟨𝑥, 𝑧⟩ ∈ ∅
4 df-br 3898 . . . . . . 7 (𝑥𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ ∅)
53, 4mtbir 643 . . . . . 6 ¬ 𝑥𝑧
65intnanr 898 . . . . 5 ¬ (𝑥𝑧𝑧𝐴𝑦)
76nex 1459 . . . 4 ¬ ∃𝑧(𝑥𝑧𝑧𝐴𝑦)
8 vex 2661 . . . . 5 𝑥 ∈ V
9 vex 2661 . . . . 5 𝑦 ∈ V
108, 9opelco 4679 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥𝑧𝑧𝐴𝑦))
117, 10mtbir 643 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
12 noel 3335 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
1311, 122false 673 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
141, 2, 13eqrelriiv 4601 1 (𝐴 ∘ ∅) = ∅
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1314  wex 1451  wcel 1463  c0 3331  cop 3498   class class class wbr 3897  ccom 4511
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-co 4516
This theorem is referenced by:  co01  5021
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