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Mirrors > Home > MPE Home > Th. List > frgra0v | Structured version Visualization version GIF version |
Description: Any graph with no vertex is a friendship graph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 4-Oct-2017.) |
Ref | Expression |
---|---|
frgra0v | ⊢ (∅ FriendGrph 𝐸 ↔ 𝐸 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frisusgra 26519 | . . 3 ⊢ (∅ FriendGrph 𝐸 → ∅ USGrph 𝐸) | |
2 | usgra0v 25900 | . . 3 ⊢ (∅ USGrph 𝐸 ↔ 𝐸 = ∅) | |
3 | 1, 2 | sylib 207 | . 2 ⊢ (∅ FriendGrph 𝐸 → 𝐸 = ∅) |
4 | 2 | biimpri 217 | . . 3 ⊢ (𝐸 = ∅ → ∅ USGrph 𝐸) |
5 | ral0 4028 | . . . 4 ⊢ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝐸 = ∅ → ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸) |
7 | 0ex 4718 | . . . 4 ⊢ ∅ ∈ V | |
8 | eleq1 2676 | . . . . 5 ⊢ (𝐸 = ∅ → (𝐸 ∈ V ↔ ∅ ∈ V)) | |
9 | 7, 8 | mpbiri 247 | . . . 4 ⊢ (𝐸 = ∅ → 𝐸 ∈ V) |
10 | isfrgra 26517 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝐸 ∈ V) → (∅ FriendGrph 𝐸 ↔ (∅ USGrph 𝐸 ∧ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))) | |
11 | 7, 9, 10 | sylancr 694 | . . 3 ⊢ (𝐸 = ∅ → (∅ FriendGrph 𝐸 ↔ (∅ USGrph 𝐸 ∧ ∀𝑘 ∈ ∅ ∀𝑙 ∈ (∅ ∖ {𝑘})∃!𝑥 ∈ ∅ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))) |
12 | 4, 6, 11 | mpbir2and 959 | . 2 ⊢ (𝐸 = ∅ → ∅ FriendGrph 𝐸) |
13 | 3, 12 | impbii 198 | 1 ⊢ (∅ FriendGrph 𝐸 ↔ 𝐸 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃!wreu 2898 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 {csn 4125 {cpr 4127 class class class wbr 4583 ran crn 5039 USGrph cusg 25859 FriendGrph cfrgra 26515 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 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-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 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-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-usgra 25862 df-frgra 26516 |
This theorem is referenced by: frgra0 26521 |
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