Theorem relsn 4854

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 4755	. 2 ⊢ (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2		relsn.1	. . 3 ⊢ 𝐴 ∈ V
3	2	snss 3828	. 2 ⊢ (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V))
4	1, 3	bitr4i 187	1 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))

Syntax hints:   ↔ wb 105   ∈ wcel 2203  Vcvv 2812   ⊆ wss 3210  {csn 3688   × cxp 4746  Rel wrel 4753

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-in 3216  df-ss 3223  df-sn 3694  df-rel 4755

This theorem is referenced by:  relsnop  4855  relsn2m  5232