Theorem relsn 4833

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 4734	. 2 ⊢ (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2		relsn.1	. . 3 ⊢ 𝐴 ∈ V
3	2	snss 3809	. 2 ⊢ (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V))
4	1, 3	bitr4i 187	1 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))

Syntax hints:   ↔ wb 105   ∈ wcel 2201  Vcvv 2801   ⊆ wss 3199  {csn 3670   × cxp 4725  Rel wrel 4732

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-v 2803  df-in 3205  df-ss 3212  df-sn 3676  df-rel 4734

This theorem is referenced by:  relsnop  4834  relsn2m  5209