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Mirrors > Home > MPE Home > Th. List > griedg0ssusgr | Structured version Visualization version GIF version |
Description: The class of all simple graphs is a superclass of the class of empty graphs represented as ordered pairs. (Contributed by AV, 27-Dec-2020.) |
Ref | Expression |
---|---|
griedg0prc.u | ⊢ 𝑈 = {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅} |
Ref | Expression |
---|---|
griedg0ssusgr | ⊢ 𝑈 ⊆ USGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | griedg0prc.u | . . . . 5 ⊢ 𝑈 = {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅} | |
2 | 1 | eleq2i 2904 | . . . 4 ⊢ (𝑔 ∈ 𝑈 ↔ 𝑔 ∈ {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅}) |
3 | elopab 5413 | . . . 4 ⊢ (𝑔 ∈ {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅} ↔ ∃𝑣∃𝑒(𝑔 = 〈𝑣, 𝑒〉 ∧ 𝑒:∅⟶∅)) | |
4 | 2, 3 | bitri 277 | . . 3 ⊢ (𝑔 ∈ 𝑈 ↔ ∃𝑣∃𝑒(𝑔 = 〈𝑣, 𝑒〉 ∧ 𝑒:∅⟶∅)) |
5 | opex 5355 | . . . . . . . 8 ⊢ 〈𝑣, 𝑒〉 ∈ V | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (𝑒:∅⟶∅ → 〈𝑣, 𝑒〉 ∈ V) |
7 | vex 3497 | . . . . . . . . 9 ⊢ 𝑣 ∈ V | |
8 | vex 3497 | . . . . . . . . 9 ⊢ 𝑒 ∈ V | |
9 | 7, 8 | opiedgfvi 26794 | . . . . . . . 8 ⊢ (iEdg‘〈𝑣, 𝑒〉) = 𝑒 |
10 | f0bi 6561 | . . . . . . . . 9 ⊢ (𝑒:∅⟶∅ ↔ 𝑒 = ∅) | |
11 | 10 | biimpi 218 | . . . . . . . 8 ⊢ (𝑒:∅⟶∅ → 𝑒 = ∅) |
12 | 9, 11 | syl5eq 2868 | . . . . . . 7 ⊢ (𝑒:∅⟶∅ → (iEdg‘〈𝑣, 𝑒〉) = ∅) |
13 | 6, 12 | usgr0e 27017 | . . . . . 6 ⊢ (𝑒:∅⟶∅ → 〈𝑣, 𝑒〉 ∈ USGraph) |
14 | 13 | adantl 484 | . . . . 5 ⊢ ((𝑔 = 〈𝑣, 𝑒〉 ∧ 𝑒:∅⟶∅) → 〈𝑣, 𝑒〉 ∈ USGraph) |
15 | eleq1 2900 | . . . . . 6 ⊢ (𝑔 = 〈𝑣, 𝑒〉 → (𝑔 ∈ USGraph ↔ 〈𝑣, 𝑒〉 ∈ USGraph)) | |
16 | 15 | adantr 483 | . . . . 5 ⊢ ((𝑔 = 〈𝑣, 𝑒〉 ∧ 𝑒:∅⟶∅) → (𝑔 ∈ USGraph ↔ 〈𝑣, 𝑒〉 ∈ USGraph)) |
17 | 14, 16 | mpbird 259 | . . . 4 ⊢ ((𝑔 = 〈𝑣, 𝑒〉 ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph) |
18 | 17 | exlimivv 1929 | . . 3 ⊢ (∃𝑣∃𝑒(𝑔 = 〈𝑣, 𝑒〉 ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph) |
19 | 4, 18 | sylbi 219 | . 2 ⊢ (𝑔 ∈ 𝑈 → 𝑔 ∈ USGraph) |
20 | 19 | ssriv 3970 | 1 ⊢ 𝑈 ⊆ USGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 ∅c0 4290 〈cop 4572 {copab 5127 ⟶wf 6350 ‘cfv 6354 iEdgciedg 26781 USGraphcusgr 26933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fv 6362 df-2nd 7689 df-iedg 26783 df-usgr 26935 |
This theorem is referenced by: usgrprc 27047 |
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