Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  co02 Structured version   Visualization version   GIF version

Theorem co02 6111
 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 6095 . 2 Rel (𝐴 ∘ ∅)
2 rel0 5671 . 2 Rel ∅
3 br0 5112 . . . . . 6 ¬ 𝑥𝑧
43intnanr 488 . . . . 5 ¬ (𝑥𝑧𝑧𝐴𝑦)
54nex 1794 . . . 4 ¬ ∃𝑧(𝑥𝑧𝑧𝐴𝑦)
6 vex 3503 . . . . 5 𝑥 ∈ V
7 vex 3503 . . . . 5 𝑦 ∈ V
86, 7opelco 5741 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ∃𝑧(𝑥𝑧𝑧𝐴𝑦))
95, 8mtbir 324 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅)
10 noel 4300 . . 3 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
119, 102false 377 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 ∘ ∅) ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
121, 2, 11eqrelriiv 5662 1 (𝐴 ∘ ∅) = ∅
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 396   = wceq 1530  ∃wex 1773   ∈ wcel 2107  ∅c0 4295  ⟨cop 4570   class class class wbr 5063   ∘ ccom 5558 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-xp 5560  df-rel 5561  df-co 5563 This theorem is referenced by:  co01  6112  gsumwmhm  17993  frmdgsum  18010  frmdup1  18012  efginvrel2  18773  0frgp  18825  evl1fval  20407  utop2nei  22774  tngds  23172  tocycf  30673  tocyc01  30674  mrsub0  32647  dfpo2  32875  cononrel1  39819
 Copyright terms: Public domain W3C validator