Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > umgr2v2evtxel | Structured version Visualization version GIF version |
Description: A vertex in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.) |
Ref | Expression |
---|---|
umgr2v2evtx.g | ⊢ 𝐺 = 〈𝑉, {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}〉 |
Ref | Expression |
---|---|
umgr2v2evtxel | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ (Vtx‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgr2v2evtx.g | . . 3 ⊢ 𝐺 = 〈𝑉, {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}〉 | |
2 | 1 | umgr2v2evtx 27869 | . 2 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉) |
3 | eqcom 2746 | . . . . 5 ⊢ ((Vtx‘𝐺) = 𝑉 ↔ 𝑉 = (Vtx‘𝐺)) | |
4 | 3 | biimpi 215 | . . . 4 ⊢ ((Vtx‘𝐺) = 𝑉 → 𝑉 = (Vtx‘𝐺)) |
5 | 4 | eleq2d 2825 | . . 3 ⊢ ((Vtx‘𝐺) = 𝑉 → (𝐴 ∈ 𝑉 ↔ 𝐴 ∈ (Vtx‘𝐺))) |
6 | 5 | biimpcd 248 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((Vtx‘𝐺) = 𝑉 → 𝐴 ∈ (Vtx‘𝐺))) |
7 | 2, 6 | mpan9 506 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ (Vtx‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 {cpr 4568 〈cop 4572 ‘cfv 6430 0cc0 10855 1c1 10856 Vtxcvtx 27347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pr 5355 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-iota 6388 df-fun 6432 df-fv 6438 df-1st 7817 df-vtx 27349 |
This theorem is referenced by: umgr2v2enb1 27874 umgr2v2evd2 27875 |
Copyright terms: Public domain | W3C validator |