Theorem relsn2OLD 5764

Description: Obsolete version of relsn2 5763 as of 12-Feb-2022. (Contributed by NM, 25-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
relsn2OLD.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
relsn2OLD	⊢ (Rel {𝐴} ↔ dom {𝐴} ≠ ∅)

Proof of Theorem relsn2OLD

Step	Hyp	Ref	Expression
1		relsn2OLD.1	. . 3 ⊢ 𝐴 ∈ V
2	1	relsn 5382	. 2 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
3		dmsnn0 5758	. 2 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
4	2, 3	bitri 264	1 ⊢ (Rel {𝐴} ↔ dom {𝐴} ≠ ∅)

Syntax hints:   ↔ wb 196   ∈ wcel 2139   ≠ wne 2932  Vcvv 3340  ∅c0 4058  {csn 4321   × cxp 5264  dom cdm 5266  Rel wrel 5271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-dm 5276

This theorem is referenced by: (None)