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Theorem co02 5810
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 5794 . 2 Rel (𝐴 ∘ ∅)
2 rel0 5399 . 2 Rel ∅
3 br0 4853 . . . . . 6 ¬ 𝑥𝑧
43intnanr 999 . . . . 5 ¬ (𝑥𝑧𝑧𝐴𝑦)
54nex 1880 . . . 4 ¬ ∃𝑧(𝑥𝑧𝑧𝐴𝑦)
6 vex 3343 . . . . 5 𝑥 ∈ V
7 vex 3343 . . . . 5 𝑦 ∈ V
86, 7opelco 5449 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥𝑧𝑧𝐴𝑦))
95, 8mtbir 312 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10 noel 4062 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
119, 102false 364 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
121, 2, 11eqrelriiv 5371 1 (𝐴 ∘ ∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wex 1853  wcel 2139  c0 4058  cop 4327   class class class wbr 4804  ccom 5270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-co 5275
This theorem is referenced by:  co01  5811  gsumwmhm  17583  frmdgsum  17600  frmdup1  17602  efginvrel2  18340  0frgp  18392  evl1fval  19894  utop2nei  22255  tngds  22653  mrsub0  31720  dfpo2  31952  cononrel1  38402
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