Theorem relsn 4604

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 4506	. 2 ⊢ (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2		relsn.1	. . 3 ⊢ 𝐴 ∈ V
3	2	snss 3615	. 2 ⊢ (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V))
4	1, 3	bitr4i 186	1 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))

Syntax hints:   ↔ wb 104   ∈ wcel 1463  Vcvv 2657   ⊆ wss 3037  {csn 3493   × cxp 4497  Rel wrel 4504

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-in 3043  df-ss 3050  df-sn 3499  df-rel 4506

This theorem is referenced by:  relsnop  4605  relsn2m  4967