| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | 1egrvtxdg1.v | . . . . 5
⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) | 
| 2 | 1 | adantl 481 | . . . 4
⊢ ((𝐵 = 𝐷 ∧ 𝜑) → (Vtx‘𝐺) = 𝑉) | 
| 3 |  | 1egrvtxdg1.a | . . . . 5
⊢ (𝜑 → 𝐴 ∈ 𝑋) | 
| 4 | 3 | adantl 481 | . . . 4
⊢ ((𝐵 = 𝐷 ∧ 𝜑) → 𝐴 ∈ 𝑋) | 
| 5 |  | 1egrvtxdg1.b | . . . . 5
⊢ (𝜑 → 𝐵 ∈ 𝑉) | 
| 6 | 5 | adantl 481 | . . . 4
⊢ ((𝐵 = 𝐷 ∧ 𝜑) → 𝐵 ∈ 𝑉) | 
| 7 |  | 1egrvtxdg0.i | . . . . . 6
⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, {𝐵, 𝐷}〉}) | 
| 8 | 7 | adantl 481 | . . . . 5
⊢ ((𝐵 = 𝐷 ∧ 𝜑) → (iEdg‘𝐺) = {〈𝐴, {𝐵, 𝐷}〉}) | 
| 9 |  | preq2 4734 | . . . . . . . . . 10
⊢ (𝐷 = 𝐵 → {𝐵, 𝐷} = {𝐵, 𝐵}) | 
| 10 | 9 | eqcoms 2745 | . . . . . . . . 9
⊢ (𝐵 = 𝐷 → {𝐵, 𝐷} = {𝐵, 𝐵}) | 
| 11 |  | dfsn2 4639 | . . . . . . . . 9
⊢ {𝐵} = {𝐵, 𝐵} | 
| 12 | 10, 11 | eqtr4di 2795 | . . . . . . . 8
⊢ (𝐵 = 𝐷 → {𝐵, 𝐷} = {𝐵}) | 
| 13 | 12 | adantr 480 | . . . . . . 7
⊢ ((𝐵 = 𝐷 ∧ 𝜑) → {𝐵, 𝐷} = {𝐵}) | 
| 14 | 13 | opeq2d 4880 | . . . . . 6
⊢ ((𝐵 = 𝐷 ∧ 𝜑) → 〈𝐴, {𝐵, 𝐷}〉 = 〈𝐴, {𝐵}〉) | 
| 15 | 14 | sneqd 4638 | . . . . 5
⊢ ((𝐵 = 𝐷 ∧ 𝜑) → {〈𝐴, {𝐵, 𝐷}〉} = {〈𝐴, {𝐵}〉}) | 
| 16 | 8, 15 | eqtrd 2777 | . . . 4
⊢ ((𝐵 = 𝐷 ∧ 𝜑) → (iEdg‘𝐺) = {〈𝐴, {𝐵}〉}) | 
| 17 |  | 1egrvtxdg1.c | . . . . . . 7
⊢ (𝜑 → 𝐶 ∈ 𝑉) | 
| 18 |  | 1egrvtxdg1.n | . . . . . . . 8
⊢ (𝜑 → 𝐵 ≠ 𝐶) | 
| 19 | 18 | necomd 2996 | . . . . . . 7
⊢ (𝜑 → 𝐶 ≠ 𝐵) | 
| 20 | 17, 19 | jca 511 | . . . . . 6
⊢ (𝜑 → (𝐶 ∈ 𝑉 ∧ 𝐶 ≠ 𝐵)) | 
| 21 |  | eldifsn 4786 | . . . . . 6
⊢ (𝐶 ∈ (𝑉 ∖ {𝐵}) ↔ (𝐶 ∈ 𝑉 ∧ 𝐶 ≠ 𝐵)) | 
| 22 | 20, 21 | sylibr 234 | . . . . 5
⊢ (𝜑 → 𝐶 ∈ (𝑉 ∖ {𝐵})) | 
| 23 | 22 | adantl 481 | . . . 4
⊢ ((𝐵 = 𝐷 ∧ 𝜑) → 𝐶 ∈ (𝑉 ∖ {𝐵})) | 
| 24 | 2, 4, 6, 16, 23 | 1loopgrvd0 29522 | . . 3
⊢ ((𝐵 = 𝐷 ∧ 𝜑) → ((VtxDeg‘𝐺)‘𝐶) = 0) | 
| 25 | 24 | ex 412 | . 2
⊢ (𝐵 = 𝐷 → (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 0)) | 
| 26 |  | necom 2994 | . . . . . . . . . 10
⊢ (𝐵 ≠ 𝐶 ↔ 𝐶 ≠ 𝐵) | 
| 27 |  | df-ne 2941 | . . . . . . . . . 10
⊢ (𝐶 ≠ 𝐵 ↔ ¬ 𝐶 = 𝐵) | 
| 28 | 26, 27 | sylbb 219 | . . . . . . . . 9
⊢ (𝐵 ≠ 𝐶 → ¬ 𝐶 = 𝐵) | 
| 29 | 18, 28 | syl 17 | . . . . . . . 8
⊢ (𝜑 → ¬ 𝐶 = 𝐵) | 
| 30 |  | 1egrvtxdg0.n | . . . . . . . . 9
⊢ (𝜑 → 𝐶 ≠ 𝐷) | 
| 31 | 30 | neneqd 2945 | . . . . . . . 8
⊢ (𝜑 → ¬ 𝐶 = 𝐷) | 
| 32 | 29, 31 | jca 511 | . . . . . . 7
⊢ (𝜑 → (¬ 𝐶 = 𝐵 ∧ ¬ 𝐶 = 𝐷)) | 
| 33 | 32 | adantl 481 | . . . . . 6
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → (¬ 𝐶 = 𝐵 ∧ ¬ 𝐶 = 𝐷)) | 
| 34 |  | ioran 986 | . . . . . 6
⊢ (¬
(𝐶 = 𝐵 ∨ 𝐶 = 𝐷) ↔ (¬ 𝐶 = 𝐵 ∧ ¬ 𝐶 = 𝐷)) | 
| 35 | 33, 34 | sylibr 234 | . . . . 5
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → ¬ (𝐶 = 𝐵 ∨ 𝐶 = 𝐷)) | 
| 36 |  | edgval 29066 | . . . . . . . . 9
⊢
(Edg‘𝐺) = ran
(iEdg‘𝐺) | 
| 37 | 7 | rneqd 5949 | . . . . . . . . . 10
⊢ (𝜑 → ran (iEdg‘𝐺) = ran {〈𝐴, {𝐵, 𝐷}〉}) | 
| 38 |  | rnsnopg 6241 | . . . . . . . . . . 11
⊢ (𝐴 ∈ 𝑋 → ran {〈𝐴, {𝐵, 𝐷}〉} = {{𝐵, 𝐷}}) | 
| 39 | 3, 38 | syl 17 | . . . . . . . . . 10
⊢ (𝜑 → ran {〈𝐴, {𝐵, 𝐷}〉} = {{𝐵, 𝐷}}) | 
| 40 | 37, 39 | eqtrd 2777 | . . . . . . . . 9
⊢ (𝜑 → ran (iEdg‘𝐺) = {{𝐵, 𝐷}}) | 
| 41 | 36, 40 | eqtrid 2789 | . . . . . . . 8
⊢ (𝜑 → (Edg‘𝐺) = {{𝐵, 𝐷}}) | 
| 42 | 41 | adantl 481 | . . . . . . 7
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → (Edg‘𝐺) = {{𝐵, 𝐷}}) | 
| 43 | 42 | rexeqdv 3327 | . . . . . 6
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → (∃𝑒 ∈ (Edg‘𝐺)𝐶 ∈ 𝑒 ↔ ∃𝑒 ∈ {{𝐵, 𝐷}}𝐶 ∈ 𝑒)) | 
| 44 |  | prex 5437 | . . . . . . 7
⊢ {𝐵, 𝐷} ∈ V | 
| 45 |  | eleq2 2830 | . . . . . . . 8
⊢ (𝑒 = {𝐵, 𝐷} → (𝐶 ∈ 𝑒 ↔ 𝐶 ∈ {𝐵, 𝐷})) | 
| 46 | 45 | rexsng 4676 | . . . . . . 7
⊢ ({𝐵, 𝐷} ∈ V → (∃𝑒 ∈ {{𝐵, 𝐷}}𝐶 ∈ 𝑒 ↔ 𝐶 ∈ {𝐵, 𝐷})) | 
| 47 | 44, 46 | mp1i 13 | . . . . . 6
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → (∃𝑒 ∈ {{𝐵, 𝐷}}𝐶 ∈ 𝑒 ↔ 𝐶 ∈ {𝐵, 𝐷})) | 
| 48 |  | elprg 4648 | . . . . . . . 8
⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ {𝐵, 𝐷} ↔ (𝐶 = 𝐵 ∨ 𝐶 = 𝐷))) | 
| 49 | 17, 48 | syl 17 | . . . . . . 7
⊢ (𝜑 → (𝐶 ∈ {𝐵, 𝐷} ↔ (𝐶 = 𝐵 ∨ 𝐶 = 𝐷))) | 
| 50 | 49 | adantl 481 | . . . . . 6
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → (𝐶 ∈ {𝐵, 𝐷} ↔ (𝐶 = 𝐵 ∨ 𝐶 = 𝐷))) | 
| 51 | 43, 47, 50 | 3bitrd 305 | . . . . 5
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → (∃𝑒 ∈ (Edg‘𝐺)𝐶 ∈ 𝑒 ↔ (𝐶 = 𝐵 ∨ 𝐶 = 𝐷))) | 
| 52 | 35, 51 | mtbird 325 | . . . 4
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → ¬ ∃𝑒 ∈ (Edg‘𝐺)𝐶 ∈ 𝑒) | 
| 53 |  | eqid 2737 | . . . . . 6
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) | 
| 54 | 3 | adantl 481 | . . . . . 6
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → 𝐴 ∈ 𝑋) | 
| 55 | 5, 1 | eleqtrrd 2844 | . . . . . . 7
⊢ (𝜑 → 𝐵 ∈ (Vtx‘𝐺)) | 
| 56 | 55 | adantl 481 | . . . . . 6
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → 𝐵 ∈ (Vtx‘𝐺)) | 
| 57 |  | 1egrvtxdg0.d | . . . . . . . 8
⊢ (𝜑 → 𝐷 ∈ 𝑉) | 
| 58 | 57, 1 | eleqtrrd 2844 | . . . . . . 7
⊢ (𝜑 → 𝐷 ∈ (Vtx‘𝐺)) | 
| 59 | 58 | adantl 481 | . . . . . 6
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → 𝐷 ∈ (Vtx‘𝐺)) | 
| 60 | 7 | adantl 481 | . . . . . 6
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → (iEdg‘𝐺) = {〈𝐴, {𝐵, 𝐷}〉}) | 
| 61 |  | simpl 482 | . . . . . 6
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → 𝐵 ≠ 𝐷) | 
| 62 | 53, 54, 56, 59, 60, 61 | usgr1e 29262 | . . . . 5
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → 𝐺 ∈ USGraph) | 
| 63 | 17, 1 | eleqtrrd 2844 | . . . . . 6
⊢ (𝜑 → 𝐶 ∈ (Vtx‘𝐺)) | 
| 64 | 63 | adantl 481 | . . . . 5
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → 𝐶 ∈ (Vtx‘𝐺)) | 
| 65 |  | eqid 2737 | . . . . . 6
⊢
(Edg‘𝐺) =
(Edg‘𝐺) | 
| 66 |  | eqid 2737 | . . . . . 6
⊢
(VtxDeg‘𝐺) =
(VtxDeg‘𝐺) | 
| 67 | 53, 65, 66 | vtxdusgr0edgnel 29513 | . . . . 5
⊢ ((𝐺 ∈ USGraph ∧ 𝐶 ∈ (Vtx‘𝐺)) → (((VtxDeg‘𝐺)‘𝐶) = 0 ↔ ¬ ∃𝑒 ∈ (Edg‘𝐺)𝐶 ∈ 𝑒)) | 
| 68 | 62, 64, 67 | syl2anc 584 | . . . 4
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → (((VtxDeg‘𝐺)‘𝐶) = 0 ↔ ¬ ∃𝑒 ∈ (Edg‘𝐺)𝐶 ∈ 𝑒)) | 
| 69 | 52, 68 | mpbird 257 | . . 3
⊢ ((𝐵 ≠ 𝐷 ∧ 𝜑) → ((VtxDeg‘𝐺)‘𝐶) = 0) | 
| 70 | 69 | ex 412 | . 2
⊢ (𝐵 ≠ 𝐷 → (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 0)) | 
| 71 | 25, 70 | pm2.61ine 3025 | 1
⊢ (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 0) |