Theorem relsn 2485

Description: A singleton of an ordered pair is a relation.

Hypothesis

Ref	Expression
relsn.1	⊢ A ∈ V

Assertion

Ref	Expression
relsn	⊢ Rel {⟨A, B⟩}

Proof of Theorem relsn

Step	Hyp	Ref	Expression
1		relsn.1	. . . . 5 ⊢ A ∈ V
2		opelxpi 2455	. . . . 5 ⊢ ((A ∈ V ∧ B ∈ V) → ⟨A, B⟩ ∈ (V × V))
3	1, 2	mpan 518	. . . 4 ⊢ (B ∈ V → ⟨A, B⟩ ∈ (V × V))
4	1, 1	pm3.2i 234	. . . . . 6 ⊢ (A ∈ V ∧ A ∈ V)
5	1	opelxp 2452	. . . . . 6 ⊢ (⟨A, A⟩ ∈ (V × V) ↔ (A ∈ V ∧ A ∈ V))
6	4, 5	mpbir 165	. . . . 5 ⊢ ⟨A, A⟩ ∈ (V × V)
7		opprc2 1907	. . . . . 6 ⊢ (¬ B ∈ V → ⟨A, B⟩ = ⟨A, A⟩)
8	7	eleq1d 1155	. . . . 5 ⊢ (¬ B ∈ V → (⟨A, B⟩ ∈ (V × V) ↔ ⟨A, A⟩ ∈ (V × V)))
9	6, 8	mpbiri 169	. . . 4 ⊢ (¬ B ∈ V → ⟨A, B⟩ ∈ (V × V))
10	3, 9	pm2.61i 110	. . 3 ⊢ ⟨A, B⟩ ∈ (V × V)
11		snssi 1851	. . 3 ⊢ (⟨A, B⟩ ∈ (V × V) → {⟨A, B⟩} ⊆ (V × V))
12	10, 11	ax-mp 6	. 2 ⊢ {⟨A, B⟩} ⊆ (V × V)
13		df-rel 2425	. 2 ⊢ (Rel {⟨A, B⟩} ↔ {⟨A, B⟩} ⊆ (V × V))
14	12, 13	mpbir 165	1 ⊢ Rel {⟨A, B⟩}

Syntax hints:  ¬ wn 1   ∧ wa 196   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  {csn 1808  ⟨cop 1810   × cxp 2408  Rel wrel 2415

This theorem is referenced by:  cnvsn 2636  funsn 2690  fsn 2895  fac0 4871

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-opab 2098  df-xp 2424  df-rel 2425

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