Theorem relsn 3249

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 3212	. . . . 5 ⊢ ((A ∈ V ⋀ B ∈ V) → ⟨A, B⟩ ∈ (V × V))
3	1, 2	mpan 694	. . . 4 ⊢ (B ∈ V → ⟨A, B⟩ ∈ (V × V))
4		opprc2 2495	. . . . 5 ⊢ (¬ B ∈ V → ⟨A, B⟩ = ⟨A, A⟩)
5	1	opelxp 3209	. . . . . 6 ⊢ (⟨A, A⟩ ∈ (V × V) ↔ (A ∈ V ⋀ A ∈ V))
6	5, 1, 1	mpbir2an 729	. . . . 5 ⊢ ⟨A, A⟩ ∈ (V × V)
7	4, 6	syl6eqel 1553	. . . 4 ⊢ (¬ B ∈ V → ⟨A, B⟩ ∈ (V × V))
8	3, 7	pm2.61i 126	. . 3 ⊢ ⟨A, B⟩ ∈ (V × V)
9		snssi 2462	. . 3 ⊢ (⟨A, B⟩ ∈ (V × V) → {⟨A, B⟩} ⊆ (V × V))
10	8, 9	ax-mp 7	. 2 ⊢ {⟨A, B⟩} ⊆ (V × V)
11		df-rel 3180	. 2 ⊢ (Rel {⟨A, B⟩} ↔ {⟨A, B⟩} ⊆ (V × V))
12	10, 11	mpbir 190	1 ⊢ Rel {⟨A, B⟩}

Syntax hints:  ¬ wn 2   ∈ wcel 956  Vcvv 1807   ⊆ wss 2043  {csn 2405  ⟨cop 2407   × cxp 3163  Rel wrel 3170

This theorem is referenced by:  cnvsn 3441  funsn 3535  fsn 3825

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-opab 2662  df-xp 3179  df-rel 3180

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