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Mirrors > Home > MPE Home > Th. List > Mathboxes > usgr0 | Structured version Visualization version GIF version |
Description: The null graph represented by an empty set is a simple graph. (Contributed by AV, 16-Oct-2020.) |
Ref | Expression |
---|---|
usgr0 | ⊢ ∅ ∈ USGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f10 6081 | . . 3 ⊢ ∅:∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2} | |
2 | dm0 5260 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | f1eq2 6010 | . . . 4 ⊢ (dom ∅ = ∅ → (∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ ∅:∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2})) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2} ↔ ∅:∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2}) |
5 | 1, 4 | mpbir 220 | . 2 ⊢ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2} |
6 | 0ex 4718 | . . 3 ⊢ ∅ ∈ V | |
7 | vtxval0 25714 | . . . . 5 ⊢ (Vtx‘∅) = ∅ | |
8 | 7 | eqcomi 2619 | . . . 4 ⊢ ∅ = (Vtx‘∅) |
9 | iedgval0 25715 | . . . . 5 ⊢ (iEdg‘∅) = ∅ | |
10 | 9 | eqcomi 2619 | . . . 4 ⊢ ∅ = (iEdg‘∅) |
11 | 8, 10 | isusgr 40383 | . . 3 ⊢ (∅ ∈ V → (∅ ∈ USGraph ↔ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2})) |
12 | 6, 11 | ax-mp 5 | . 2 ⊢ (∅ ∈ USGraph ↔ ∅:dom ∅–1-1→{𝑥 ∈ (𝒫 ∅ ∖ {∅}) ∣ (#‘𝑥) = 2}) |
13 | 5, 12 | mpbir 220 | 1 ⊢ ∅ ∈ USGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ∖ cdif 3537 ∅c0 3874 𝒫 cpw 4108 {csn 4125 dom cdm 5038 –1-1→wf1 5801 ‘cfv 5804 2c2 10947 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 USGraph cusgr 40379 |
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-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 |
This theorem is referenced by: cusgr0 40648 frgr0 41436 |
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