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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 2906 | . . . 4 ⊢ (𝑔 ∈ 𝑈 ↔ 𝑔 ∈ {〈𝑣, 𝑒〉 ∣ 𝑒:∅⟶∅}) |
3 | elopab 5416 | . . . 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 5358 | . . . . . . . 8 ⊢ 〈𝑣, 𝑒〉 ∈ V | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (𝑒:∅⟶∅ → 〈𝑣, 𝑒〉 ∈ V) |
7 | vex 3499 | . . . . . . . . 9 ⊢ 𝑣 ∈ V | |
8 | vex 3499 | . . . . . . . . 9 ⊢ 𝑒 ∈ V | |
9 | 7, 8 | opiedgfvi 26797 | . . . . . . . 8 ⊢ (iEdg‘〈𝑣, 𝑒〉) = 𝑒 |
10 | f0bi 6564 | . . . . . . . . 9 ⊢ (𝑒:∅⟶∅ ↔ 𝑒 = ∅) | |
11 | 10 | biimpi 218 | . . . . . . . 8 ⊢ (𝑒:∅⟶∅ → 𝑒 = ∅) |
12 | 9, 11 | syl5eq 2870 | . . . . . . 7 ⊢ (𝑒:∅⟶∅ → (iEdg‘〈𝑣, 𝑒〉) = ∅) |
13 | 6, 12 | usgr0e 27020 | . . . . . 6 ⊢ (𝑒:∅⟶∅ → 〈𝑣, 𝑒〉 ∈ USGraph) |
14 | 13 | adantl 484 | . . . . 5 ⊢ ((𝑔 = 〈𝑣, 𝑒〉 ∧ 𝑒:∅⟶∅) → 〈𝑣, 𝑒〉 ∈ USGraph) |
15 | eleq1 2902 | . . . . . 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 1933 | . . 3 ⊢ (∃𝑣∃𝑒(𝑔 = 〈𝑣, 𝑒〉 ∧ 𝑒:∅⟶∅) → 𝑔 ∈ USGraph) |
19 | 4, 18 | sylbi 219 | . 2 ⊢ (𝑔 ∈ 𝑈 → 𝑔 ∈ USGraph) |
20 | 19 | ssriv 3973 | 1 ⊢ 𝑈 ⊆ USGraph |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 〈cop 4575 {copab 5130 ⟶wf 6353 ‘cfv 6357 iEdgciedg 26784 USGraphcusgr 26936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fv 6365 df-2nd 7692 df-iedg 26786 df-usgr 26938 |
This theorem is referenced by: usgrprc 27050 |
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