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Mirrors > Home > MPE Home > Th. List > cusgra1v | Structured version Visualization version GIF version |
Description: A graph with one vertex (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.) |
Ref | Expression |
---|---|
cusgra1v | ⊢ {𝐴} ComplUSGrph ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 4835 | . . . 4 ⊢ {𝐴} ∈ V | |
2 | usgra0 25899 | . . . 4 ⊢ ({𝐴} ∈ V → {𝐴} USGrph ∅) | |
3 | 1, 2 | mp1i 13 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} USGrph ∅) |
4 | ral0 4028 | . . . 4 ⊢ ∀𝑛 ∈ ∅ {𝑛, 𝐴} ∈ ran ∅ | |
5 | sneq 4135 | . . . . . . . 8 ⊢ (𝑘 = 𝐴 → {𝑘} = {𝐴}) | |
6 | 5 | difeq2d 3690 | . . . . . . 7 ⊢ (𝑘 = 𝐴 → ({𝐴} ∖ {𝑘}) = ({𝐴} ∖ {𝐴})) |
7 | difid 3902 | . . . . . . 7 ⊢ ({𝐴} ∖ {𝐴}) = ∅ | |
8 | 6, 7 | syl6eq 2660 | . . . . . 6 ⊢ (𝑘 = 𝐴 → ({𝐴} ∖ {𝑘}) = ∅) |
9 | preq2 4213 | . . . . . . 7 ⊢ (𝑘 = 𝐴 → {𝑛, 𝑘} = {𝑛, 𝐴}) | |
10 | 9 | eleq1d 2672 | . . . . . 6 ⊢ (𝑘 = 𝐴 → ({𝑛, 𝑘} ∈ ran ∅ ↔ {𝑛, 𝐴} ∈ ran ∅)) |
11 | 8, 10 | raleqbidv 3129 | . . . . 5 ⊢ (𝑘 = 𝐴 → (∀𝑛 ∈ ({𝐴} ∖ {𝑘}){𝑛, 𝑘} ∈ ran ∅ ↔ ∀𝑛 ∈ ∅ {𝑛, 𝐴} ∈ ran ∅)) |
12 | 11 | ralsng 4165 | . . . 4 ⊢ (𝐴 ∈ V → (∀𝑘 ∈ {𝐴}∀𝑛 ∈ ({𝐴} ∖ {𝑘}){𝑛, 𝑘} ∈ ran ∅ ↔ ∀𝑛 ∈ ∅ {𝑛, 𝐴} ∈ ran ∅)) |
13 | 4, 12 | mpbiri 247 | . . 3 ⊢ (𝐴 ∈ V → ∀𝑘 ∈ {𝐴}∀𝑛 ∈ ({𝐴} ∖ {𝑘}){𝑛, 𝑘} ∈ ran ∅) |
14 | 0ex 4718 | . . . 4 ⊢ ∅ ∈ V | |
15 | iscusgra 25985 | . . . 4 ⊢ (({𝐴} ∈ V ∧ ∅ ∈ V) → ({𝐴} ComplUSGrph ∅ ↔ ({𝐴} USGrph ∅ ∧ ∀𝑘 ∈ {𝐴}∀𝑛 ∈ ({𝐴} ∖ {𝑘}){𝑛, 𝑘} ∈ ran ∅))) | |
16 | 1, 14, 15 | mp2an 704 | . . 3 ⊢ ({𝐴} ComplUSGrph ∅ ↔ ({𝐴} USGrph ∅ ∧ ∀𝑘 ∈ {𝐴}∀𝑛 ∈ ({𝐴} ∖ {𝑘}){𝑛, 𝑘} ∈ ran ∅)) |
17 | 3, 13, 16 | sylanbrc 695 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ComplUSGrph ∅) |
18 | snprc 4197 | . . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
19 | cusgra0v 25989 | . . . 4 ⊢ ∅ ComplUSGrph ∅ | |
20 | breq1 4586 | . . . 4 ⊢ ({𝐴} = ∅ → ({𝐴} ComplUSGrph ∅ ↔ ∅ ComplUSGrph ∅)) | |
21 | 19, 20 | mpbiri 247 | . . 3 ⊢ ({𝐴} = ∅ → {𝐴} ComplUSGrph ∅) |
22 | 18, 21 | sylbi 206 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} ComplUSGrph ∅) |
23 | 17, 22 | pm2.61i 175 | 1 ⊢ {𝐴} ComplUSGrph ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∖ cdif 3537 ∅c0 3874 {csn 4125 {cpr 4127 class class class wbr 4583 ran crn 5039 USGrph cusg 25859 ComplUSGrph ccusgra 25947 |
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-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-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-cusgra 25950 |
This theorem is referenced by: (None) |
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