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Theorem co02 4761
 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 4746 . 2 Rel (A ∘ ∅)
2 rel0 4389 . 2 Rel ∅
3 noel 3205 . . . . . . 7 ¬ ⟨x, z
4 df-br 3739 . . . . . . 7 (xz ↔ ⟨x, z ∅)
53, 4mtbir 583 . . . . . 6 ¬ xz
65intnanr 827 . . . . 5 ¬ (xz zAy)
76nex 1370 . . . 4 ¬ z(xz zAy)
8 vex 2538 . . . . 5 x V
9 vex 2538 . . . . 5 y V
108, 9opelco 4434 . . . 4 (⟨x, y (A ∘ ∅) ↔ z(xz zAy))
117, 10mtbir 583 . . 3 ¬ ⟨x, y (A ∘ ∅)
12 noel 3205 . . 3 ¬ ⟨x, y
1311, 122false 604 . 2 (⟨x, y (A ∘ ∅) ↔ ⟨x, y ∅)
141, 2, 13eqrelriiv 4361 1 (A ∘ ∅) = ∅
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ∅c0 3201  ⟨cop 3353   class class class wbr 3738   ∘ ccom 4276 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-co 4281 This theorem is referenced by:  co01  4762
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