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Mirrors > Home > MPE Home > Th. List > elusuhgra | Structured version Visualization version GIF version |
Description: An undirected simple graph without loops is an undirected hypergraph. (Contributed by AV, 9-Jan-2020.) |
Ref | Expression |
---|---|
elusuhgra | ⊢ (𝐺 ∈ USGrph → 𝐺 ∈ UHGrph ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relusgra 25864 | . . 3 ⊢ Rel USGrph | |
2 | 1st2nd 7105 | . . 3 ⊢ ((Rel USGrph ∧ 𝐺 ∈ USGrph ) → 𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉) | |
3 | 1, 2 | mpan 702 | . 2 ⊢ (𝐺 ∈ USGrph → 𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉) |
4 | usisuhgra 25896 | . . . 4 ⊢ ((1st ‘𝐺) USGrph (2nd ‘𝐺) → (1st ‘𝐺) UHGrph (2nd ‘𝐺)) | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉 → ((1st ‘𝐺) USGrph (2nd ‘𝐺) → (1st ‘𝐺) UHGrph (2nd ‘𝐺))) |
6 | eleq1 2676 | . . . 4 ⊢ (𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉 → (𝐺 ∈ USGrph ↔ 〈(1st ‘𝐺), (2nd ‘𝐺)〉 ∈ USGrph )) | |
7 | df-br 4584 | . . . 4 ⊢ ((1st ‘𝐺) USGrph (2nd ‘𝐺) ↔ 〈(1st ‘𝐺), (2nd ‘𝐺)〉 ∈ USGrph ) | |
8 | 6, 7 | syl6bbr 277 | . . 3 ⊢ (𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉 → (𝐺 ∈ USGrph ↔ (1st ‘𝐺) USGrph (2nd ‘𝐺))) |
9 | eleq1 2676 | . . . 4 ⊢ (𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉 → (𝐺 ∈ UHGrph ↔ 〈(1st ‘𝐺), (2nd ‘𝐺)〉 ∈ UHGrph )) | |
10 | df-br 4584 | . . . 4 ⊢ ((1st ‘𝐺) UHGrph (2nd ‘𝐺) ↔ 〈(1st ‘𝐺), (2nd ‘𝐺)〉 ∈ UHGrph ) | |
11 | 9, 10 | syl6bbr 277 | . . 3 ⊢ (𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉 → (𝐺 ∈ UHGrph ↔ (1st ‘𝐺) UHGrph (2nd ‘𝐺))) |
12 | 5, 8, 11 | 3imtr4d 282 | . 2 ⊢ (𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉 → (𝐺 ∈ USGrph → 𝐺 ∈ UHGrph )) |
13 | 3, 12 | mpcom 37 | 1 ⊢ (𝐺 ∈ USGrph → 𝐺 ∈ UHGrph ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 Rel wrel 5043 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 UHGrph cuhg 25819 USGrph cusg 25859 |
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 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 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-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-2 10956 df-uhgra 25821 df-umgra 25842 df-uslgra 25861 df-usgra 25862 |
This theorem is referenced by: 0eusgraiff0rgracl 26468 |
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