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Mirrors > Home > MPE Home > Th. List > iedgval0 | Structured version Visualization version GIF version |
Description: Degenerated case 1 for edges: The set of indexed edges of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.) |
Ref | Expression |
---|---|
iedgval0 | ⊢ (iEdg‘∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelxp 5583 | . . 3 ⊢ ¬ ∅ ∈ (V × V) | |
2 | 1 | iffalsei 4476 | . 2 ⊢ if(∅ ∈ (V × V), (2nd ‘∅), (.ef‘∅)) = (.ef‘∅) |
3 | iedgval 26780 | . 2 ⊢ (iEdg‘∅) = if(∅ ∈ (V × V), (2nd ‘∅), (.ef‘∅)) | |
4 | df-edgf 26769 | . . 3 ⊢ .ef = Slot ;18 | |
5 | 4 | str0 16529 | . 2 ⊢ ∅ = (.ef‘∅) |
6 | 2, 3, 5 | 3eqtr4i 2854 | 1 ⊢ (iEdg‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 ifcif 4466 × cxp 5547 ‘cfv 6349 2nd c2nd 7682 1c1 10532 8c8 11692 ;cdc 12092 .efcedgf 26768 iEdgciedg 26776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-slot 16481 df-edgf 26769 df-iedg 26778 |
This theorem is referenced by: uhgr0 26852 usgr0 27019 0grsubgr 27054 0grrusgr 27355 |
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