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Theorem co02 5548
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 5532 . 2 Rel (𝐴 ∘ ∅)
2 rel0 5151 . 2 Rel ∅
3 br0 4621 . . . . . 6 ¬ 𝑥𝑧
43intnanr 951 . . . . 5 ¬ (𝑥𝑧𝑧𝐴𝑦)
54nex 1720 . . . 4 ¬ ∃𝑧(𝑥𝑧𝑧𝐴𝑦)
6 vex 3171 . . . . 5 𝑥 ∈ V
7 vex 3171 . . . . 5 𝑦 ∈ V
86, 7opelco 5199 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥𝑧𝑧𝐴𝑦))
95, 8mtbir 311 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10 noel 3873 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
119, 102false 363 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
121, 2, 11eqrelriiv 5122 1 (𝐴 ∘ ∅) = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wex 1694  wcel 1975  c0 3869  cop 4126   class class class wbr 4573  ccom 5028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-opab 4634  df-xp 5030  df-rel 5031  df-co 5033
This theorem is referenced by:  co01  5549  gsumwmhm  17147  frmdgsum  17164  frmdup1  17166  efginvrel2  17905  0frgp  17957  evl1fval  19455  utop2nei  21802  tngds  22196  mrsub0  30469  dfpo2  30700  cononrel1  36718
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