Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2829 | . . . 4 ⊢ (𝑔 ∈ 𝑈 ↔ 𝑔 ∈ {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅}) |
3 | elopab 5408 | . . . 4 ⊢ (𝑔 ∈ {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅} ↔ ∃𝑣∃𝑒(𝑔 = 〈𝑣, 𝑒〉 ∧ 𝑒:∅⟶∅)) | |
4 | 2, 3 | bitri 278 | . . 3 ⊢ (𝑔 ∈ 𝑈 ↔ ∃𝑣∃𝑒(𝑔 = 〈𝑣, 𝑒〉 ∧ 𝑒:∅⟶∅)) |
5 | opex 5348 | . . . . . . . 8 ⊢ 〈𝑣, 𝑒〉 ∈ V | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (𝑒:∅⟶∅ → 〈𝑣, 𝑒〉 ∈ V) |
7 | vex 3412 | . . . . . . . . 9 ⊢ 𝑣 ∈ V | |
8 | vex 3412 | . . . . . . . . 9 ⊢ 𝑒 ∈ V | |
9 | 7, 8 | opiedgfvi 27101 | . . . . . . . 8 ⊢ (iEdg‘〈𝑣, 𝑒〉) = 𝑒 |
10 | f0bi 6602 | . . . . . . . . 9 ⊢ (𝑒:∅⟶∅ ↔ 𝑒 = ∅) | |
11 | 10 | biimpi 219 | . . . . . . . 8 ⊢ (𝑒:∅⟶∅ → 𝑒 = ∅) |
12 | 9, 11 | syl5eq 2790 | . . . . . . 7 ⊢ (𝑒:∅⟶∅ → (iEdg‘〈𝑣, 𝑒〉) = ∅) |
13 | 6, 12 | usgr0e 27324 | . . . . . 6 ⊢ (𝑒:∅⟶∅ → 〈𝑣, 𝑒〉 ∈ USGraph) |
14 | 13 | adantl 485 | . . . . 5 ⊢ ((𝑔 = 〈𝑣, 𝑒〉 ∧ 𝑒:∅⟶∅) → 〈𝑣, 𝑒〉 ∈ USGraph) |
15 | eleq1 2825 | . . . . . 6 ⊢ (𝑔 = 〈𝑣, 𝑒〉 → (𝑔 ∈ USGraph ↔ 〈𝑣, 𝑒〉 ∈ USGraph)) | |
16 | 15 | adantr 484 | . . . . 5 ⊢ ((𝑔 = 〈𝑣, 𝑒〉 ∧ 𝑒:∅⟶∅) → (𝑔 ∈ USGraph ↔ 〈𝑣, 𝑒〉 ∈ USGraph)) |
17 | 14, 16 | mpbird 260 | . . . 4 ⊢ ((𝑔 = 〈𝑣, 𝑒〉 ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph) |
18 | 17 | exlimivv 1940 | . . 3 ⊢ (∃𝑣∃𝑒(𝑔 = 〈𝑣, 𝑒〉 ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph) |
19 | 4, 18 | sylbi 220 | . 2 ⊢ (𝑔 ∈ 𝑈 → 𝑔 ∈ USGraph) |
20 | 19 | ssriv 3905 | 1 ⊢ 𝑈 ⊆ USGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2110 Vcvv 3408 ⊆ wss 3866 ∅c0 4237 〈cop 4547 {copab 5115 ⟶wf 6376 ‘cfv 6380 iEdgciedg 27088 USGraphcusgr 27240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fv 6388 df-2nd 7762 df-iedg 27090 df-usgr 27242 |
This theorem is referenced by: usgrprc 27354 |
Copyright terms: Public domain | W3C validator |