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Mirrors > Home > MPE Home > Th. List > iedgval3sn | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
vtxval3sn.a | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
iedgval3sn | ⊢ (iEdg‘{{{𝐴}}}) = {𝐴} |
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} |
Colors of variables: wff setvar class |
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) |
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