Theorem relsn 4764

Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)

Hypothesis

Ref	Expression
relsn.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
relsn	⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))

Proof of Theorem relsn

Step	Hyp	Ref	Expression
1		df-rel 4666	. 2 ⊢ (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2		relsn.1	. . 3 ⊢ 𝐴 ∈ V
3	2	snss 3753	. 2 ⊢ (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V))
4	1, 3	bitr4i 187	1 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))

Syntax hints:   ↔ wb 105   ∈ wcel 2164  Vcvv 2760   ⊆ wss 3153  {csn 3618   × cxp 4657  Rel wrel 4664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-sn 3624  df-rel 4666

This theorem is referenced by:  relsnop  4765  relsn2m  5136