Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  relsnOLD Structured version   Visualization version   GIF version

Theorem relsnOLD 5366
 Description: Obsolete proof of relsn 5365 as of 12-Feb-2022. (Contributed by NM, 24-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
relsn.1 𝐴 ∈ V
Assertion
Ref Expression
relsnOLD (Rel {𝐴} ↔ 𝐴 ∈ (V × V))

Proof of Theorem relsnOLD
StepHypRef Expression
1 df-rel 5256 . 2 (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2 relsn.1 . . 3 𝐴 ∈ V
32snss 4451 . 2 (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V))
41, 3bitr4i 267 1 (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∈ wcel 2145  Vcvv 3351   ⊆ wss 3723  {csn 4316   × cxp 5247  Rel wrel 5254 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-ss 3737  df-sn 4317  df-rel 5256 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator