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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 4730 | . . . 4 ⊢ 〈𝐴, 𝐴〉 = {{𝐴}} |
3 | 2 | eqcomi 2804 | . . 3 ⊢ {{𝐴}} = 〈𝐴, 𝐴〉 |
4 | 3 | sneqi 4483 | . 2 ⊢ {{{𝐴}}} = {〈𝐴, 𝐴〉} |
5 | 1, 4 | iedgvalsnop 26510 | 1 ⊢ (iEdg‘{{{𝐴}}}) = {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ∈ wcel 2081 Vcvv 3437 {csn 4472 〈cop 4478 ‘cfv 6225 iEdgciedg 26465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-iota 6189 df-fun 6227 df-fv 6233 df-2nd 7546 df-iedg 26467 |
This theorem is referenced by: (None) |
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