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Mirrors > Home > MPE Home > Th. List > Mathboxes > frgr0 | Structured version Visualization version GIF version |
Description: The null graph (graph without vertices) is a friendship graph. (Contributed by AV, 29-Mar-2021.) |
Ref | Expression |
---|---|
frgr0 | ⊢ ∅ ∈ FriendGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgr0 40469 | . 2 ⊢ ∅ ∈ USGraph | |
2 | ral0 4028 | . 2 ⊢ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘∅) | |
3 | vtxval0 25714 | . . . 4 ⊢ (Vtx‘∅) = ∅ | |
4 | 3 | eqcomi 2619 | . . 3 ⊢ ∅ = (Vtx‘∅) |
5 | eqid 2610 | . . 3 ⊢ (Edg‘∅) = (Edg‘∅) | |
6 | 4, 5 | frgrusgrfrcond 41431 | . 2 ⊢ (∅ ∈ FriendGraph ↔ (∅ ∈ USGraph ∧ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘∅))) |
7 | 1, 2, 6 | mpbir2an 957 | 1 ⊢ ∅ ∈ FriendGraph |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ∀wral 2896 ∃!wreu 2898 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 {csn 4125 {cpr 4127 ‘cfv 5804 Vtxcvtx 25673 Edgcedga 25792 USGraph cusgr 40379 FriendGraph cfrgr 41428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fv 5812 df-slot 15699 df-base 15700 df-edgf 25668 df-vtx 25675 df-iedg 25676 df-usgr 40381 df-frgr 41429 |
This theorem is referenced by: (None) |
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