Theorem iedgval3sn 28856

Description: Degenerated case 3 for edges: The set of indexed edges of a singleton containing a singleton containing a singleton is the innermost singleton. (Contributed by AV, 24-Sep-2020.) (Avoid depending on this detail.)

Hypothesis

Ref	Expression
vtxval3sn.a	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
iedgval3sn	⊢ (iEdg‘{{{𝐴}}}) = {𝐴}

Proof of Theorem iedgval3sn

Step	Hyp	Ref	Expression
1		vtxval3sn.a	. 2 ⊢ 𝐴 ∈ V
2	1	opid 4894	. . . 4 ⊢ ⟨𝐴, 𝐴⟩ = {{𝐴}}
3	2	eqcomi 2737	. . 3 ⊢ {{𝐴}} = ⟨𝐴, 𝐴⟩
4	3	sneqi 4640	. 2 ⊢ {{{𝐴}}} = {⟨𝐴, 𝐴⟩}
5	1, 4	iedgvalsnop 28854	1 ⊢ (iEdg‘{{{𝐴}}}) = {𝐴}

Syntax hints:   = wceq 1534   ∈ wcel 2099  Vcvv 3471  {csn 4629  ⟨cop 4635  ‘cfv 6548  iEdgciedg 28809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fv 6556  df-2nd 7994  df-iedg 28811

This theorem is referenced by: (None)