Theorem co02 2663

Description: Composition with the empty set. Theorem 20 of [Suppes] p. 63.

Assertion

Ref	Expression
co02	⊢ (A ∘ ∅) = ∅

Proof of Theorem co02

Step	Hyp	Ref	Expression
1		relco 2658	. 2 ⊢ Rel (A ∘ ∅)
2		rel0 2499	. 2 ⊢ Rel ∅
3		noel 1711	. . . . . . 7 ⊢ ¬ ⟨x, z⟩ ∈ ∅
4		df-br 2063	. . . . . . 7 ⊢ (x∅z ↔ ⟨x, z⟩ ∈ ∅)
5	3, 4	mtbir 167	. . . . . 6 ⊢ ¬ x∅z
6	5	intnanr 517	. . . . 5 ⊢ ¬ (x∅z ∧ zAy)
7	6	nex 779	. . . 4 ⊢ ¬ ∃z(x∅z ∧ zAy)
8		visset 1350	. . . . 5 ⊢ x ∈ V
9		visset 1350	. . . . 5 ⊢ y ∈ V
10	8, 9	opelco 2509	. . . 4 ⊢ (⟨x, y⟩ ∈ (A ∘ ∅) ↔ ∃z(x∅z ∧ zAy))
11	7, 10	mtbir 167	. . 3 ⊢ ¬ ⟨x, y⟩ ∈ (A ∘ ∅)
12		noel 1711	. . 3 ⊢ ¬ ⟨x, y⟩ ∈ ∅
13		pm5.21 502	. . 3 ⊢ ((¬ ⟨x, y⟩ ∈ (A ∘ ∅) ∧ ¬ ⟨x, y⟩ ∈ ∅) → (⟨x, y⟩ ∈ (A ∘ ∅) ↔ ⟨x, y⟩ ∈ ∅))
14	11, 12, 13	mp2an 520	. 2 ⊢ (⟨x, y⟩ ∈ (A ∘ ∅) ↔ ⟨x, y⟩ ∈ ∅)
15	1, 2, 14	cleqreli 2484	1 ⊢ (A ∘ ∅) = ∅

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∅c0 1707  ⟨cop 1810   class class class wbr 2054   ∘ ccom 2414

This theorem is referenced by:  co01 2664

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-co 2427

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