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Theorem frgr3vlem1 29217
Description: Lemma 1 for frgr3v 29219. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.)
Hypotheses
Ref Expression
frgr3v.v 𝑉 = (Vtx‘𝐺)
frgr3v.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
frgr3vlem1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ∀𝑥𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → 𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐸,𝑦   𝑥,𝐺,𝑦   𝑥,𝑉,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝑍,𝑦

Proof of Theorem frgr3vlem1
StepHypRef Expression
1 vex 3449 . . . . . 6 𝑥 ∈ V
21eltp 4649 . . . . 5 (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶))
3 vex 3449 . . . . . . . . 9 𝑦 ∈ V
43eltp 4649 . . . . . . . 8 (𝑦 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶))
5 eqidd 2737 . . . . . . . . . . . . . . 15 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐴)
65a1i 11 . . . . . . . . . . . . . 14 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐴))
76a1i13 27 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐴))))
8 preq1 4694 . . . . . . . . . . . . . . 15 (𝑦 = 𝐴 → {𝑦, 𝐴} = {𝐴, 𝐴})
9 preq1 4694 . . . . . . . . . . . . . . 15 (𝑦 = 𝐴 → {𝑦, 𝐵} = {𝐴, 𝐵})
108, 9preq12d 4702 . . . . . . . . . . . . . 14 (𝑦 = 𝐴 → {{𝑦, 𝐴}, {𝑦, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
1110sseq1d 3975 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸))
12 eqeq2 2748 . . . . . . . . . . . . . . 15 (𝑦 = 𝐴 → (𝐴 = 𝑦𝐴 = 𝐴))
1312imbi2d 340 . . . . . . . . . . . . . 14 (𝑦 = 𝐴 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦) ↔ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐴)))
1413imbi2d 340 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐴))))
157, 11, 143imtr4d 293 . . . . . . . . . . . 12 (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦))))
16 prex 5389 . . . . . . . . . . . . . . . . . . 19 {𝐴, 𝐴} ∈ V
17 prex 5389 . . . . . . . . . . . . . . . . . . 19 {𝐴, 𝐵} ∈ V
1816, 17prss 4780 . . . . . . . . . . . . . . . . . 18 (({𝐴, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝐵} ∈ 𝐸) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸)
19 frgr3v.e . . . . . . . . . . . . . . . . . . . . . . 23 𝐸 = (Edg‘𝐺)
2019usgredgne 28154 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺 ∈ USGraph ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴𝐴)
2120adantll 712 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴𝐴)
22 df-ne 2944 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴𝐴 ↔ ¬ 𝐴 = 𝐴)
23 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . 23 𝐴 = 𝐴
2423pm2.24i 150 . . . . . . . . . . . . . . . . . . . . . 22 𝐴 = 𝐴𝐴 = 𝐵)
2522, 24sylbi 216 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝐴𝐴 = 𝐵)
2621, 25syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 = 𝐵)
2726expcom 414 . . . . . . . . . . . . . . . . . . 19 ({𝐴, 𝐴} ∈ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐴 = 𝐵))
2827adantr 481 . . . . . . . . . . . . . . . . . 18 (({𝐴, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝐵} ∈ 𝐸) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐴 = 𝐵))
2918, 28sylbir 234 . . . . . . . . . . . . . . . . 17 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐴 = 𝐵))
3029com12 32 . . . . . . . . . . . . . . . 16 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸𝐴 = 𝐵))
31303ad2ant3 1135 . . . . . . . . . . . . . . 15 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸𝐴 = 𝐵))
3231com12 32 . . . . . . . . . . . . . 14 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐵))
33322a1i 12 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐵))))
34 preq1 4694 . . . . . . . . . . . . . . 15 (𝑦 = 𝐵 → {𝑦, 𝐴} = {𝐵, 𝐴})
35 preq1 4694 . . . . . . . . . . . . . . 15 (𝑦 = 𝐵 → {𝑦, 𝐵} = {𝐵, 𝐵})
3634, 35preq12d 4702 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → {{𝑦, 𝐴}, {𝑦, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
3736sseq1d 3975 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸))
38 eqeq2 2748 . . . . . . . . . . . . . . 15 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
3938imbi2d 340 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦) ↔ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐵)))
4039imbi2d 340 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐵))))
4133, 37, 403imtr4d 293 . . . . . . . . . . . 12 (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦))))
4223pm2.24i 150 . . . . . . . . . . . . . . . . . . . . . 22 𝐴 = 𝐴𝐴 = 𝐶)
4322, 42sylbi 216 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝐴𝐴 = 𝐶)
4421, 43syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴 = 𝐶)
4544expcom 414 . . . . . . . . . . . . . . . . . . 19 ({𝐴, 𝐴} ∈ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐴 = 𝐶))
4645adantr 481 . . . . . . . . . . . . . . . . . 18 (({𝐴, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝐵} ∈ 𝐸) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐴 = 𝐶))
4718, 46sylbir 234 . . . . . . . . . . . . . . . . 17 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐴 = 𝐶))
4847com12 32 . . . . . . . . . . . . . . . 16 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸𝐴 = 𝐶))
49483ad2ant3 1135 . . . . . . . . . . . . . . 15 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸𝐴 = 𝐶))
5049com12 32 . . . . . . . . . . . . . 14 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐶))
51502a1i 12 . . . . . . . . . . . . 13 (𝑦 = 𝐶 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐶))))
52 preq1 4694 . . . . . . . . . . . . . . 15 (𝑦 = 𝐶 → {𝑦, 𝐴} = {𝐶, 𝐴})
53 preq1 4694 . . . . . . . . . . . . . . 15 (𝑦 = 𝐶 → {𝑦, 𝐵} = {𝐶, 𝐵})
5452, 53preq12d 4702 . . . . . . . . . . . . . 14 (𝑦 = 𝐶 → {{𝑦, 𝐴}, {𝑦, 𝐵}} = {{𝐶, 𝐴}, {𝐶, 𝐵}})
5554sseq1d 3975 . . . . . . . . . . . . 13 (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 ↔ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸))
56 eqeq2 2748 . . . . . . . . . . . . . . 15 (𝑦 = 𝐶 → (𝐴 = 𝑦𝐴 = 𝐶))
5756imbi2d 340 . . . . . . . . . . . . . 14 (𝑦 = 𝐶 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦) ↔ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐶)))
5857imbi2d 340 . . . . . . . . . . . . 13 (𝑦 = 𝐶 → (({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝐶))))
5951, 55, 583imtr4d 293 . . . . . . . . . . . 12 (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦))))
6015, 41, 593jaoi 1427 . . . . . . . . . . 11 ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦))))
61 preq1 4694 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → {𝑥, 𝐴} = {𝐴, 𝐴})
62 preq1 4694 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
6361, 62preq12d 4702 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
6463sseq1d 3975 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸))
65 eqeq1 2740 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
6665imbi2d 340 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦) ↔ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦)))
6764, 66imbi12d 344 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦))))
6867imbi2d 340 . . . . . . . . . . 11 (𝑥 = 𝐴 → (({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦))) ↔ ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐴 = 𝑦)))))
6960, 68syl5ibr 245 . . . . . . . . . 10 (𝑥 = 𝐴 → ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦)))))
70 prex 5389 . . . . . . . . . . . . . . . . . . 19 {𝐵, 𝐴} ∈ V
71 prex 5389 . . . . . . . . . . . . . . . . . . 19 {𝐵, 𝐵} ∈ V
7270, 71prss 4780 . . . . . . . . . . . . . . . . . 18 (({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝐵} ∈ 𝐸) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸)
7319usgredgne 28154 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺 ∈ USGraph ∧ {𝐵, 𝐵} ∈ 𝐸) → 𝐵𝐵)
7473adantll 712 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐵, 𝐵} ∈ 𝐸) → 𝐵𝐵)
75 df-ne 2944 . . . . . . . . . . . . . . . . . . . . . 22 (𝐵𝐵 ↔ ¬ 𝐵 = 𝐵)
76 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . 23 𝐵 = 𝐵
7776pm2.24i 150 . . . . . . . . . . . . . . . . . . . . . 22 𝐵 = 𝐵𝐵 = 𝐴)
7875, 77sylbi 216 . . . . . . . . . . . . . . . . . . . . 21 (𝐵𝐵𝐵 = 𝐴)
7974, 78syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐵, 𝐵} ∈ 𝐸) → 𝐵 = 𝐴)
8079expcom 414 . . . . . . . . . . . . . . . . . . 19 ({𝐵, 𝐵} ∈ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐵 = 𝐴))
8180adantl 482 . . . . . . . . . . . . . . . . . 18 (({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝐵} ∈ 𝐸) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐵 = 𝐴))
8272, 81sylbir 234 . . . . . . . . . . . . . . . . 17 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐵 = 𝐴))
8382com12 32 . . . . . . . . . . . . . . . 16 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸𝐵 = 𝐴))
84833ad2ant3 1135 . . . . . . . . . . . . . . 15 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸𝐵 = 𝐴))
8584com12 32 . . . . . . . . . . . . . 14 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐴))
86852a1i 12 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐴))))
87 eqeq2 2748 . . . . . . . . . . . . . . 15 (𝑦 = 𝐴 → (𝐵 = 𝑦𝐵 = 𝐴))
8887imbi2d 340 . . . . . . . . . . . . . 14 (𝑦 = 𝐴 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦) ↔ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐴)))
8988imbi2d 340 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐴))))
9086, 11, 893imtr4d 293 . . . . . . . . . . . 12 (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦))))
91 eqidd 2737 . . . . . . . . . . . . . . 15 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐵)
9291a1i 11 . . . . . . . . . . . . . 14 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐵))
9392a1i13 27 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐵))))
94 eqeq2 2748 . . . . . . . . . . . . . . 15 (𝑦 = 𝐵 → (𝐵 = 𝑦𝐵 = 𝐵))
9594imbi2d 340 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦) ↔ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐵)))
9695imbi2d 340 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐵))))
9793, 37, 963imtr4d 293 . . . . . . . . . . . 12 (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦))))
9876pm2.24i 150 . . . . . . . . . . . . . . . . . . . . . 22 𝐵 = 𝐵𝐵 = 𝐶)
9975, 98sylbi 216 . . . . . . . . . . . . . . . . . . . . 21 (𝐵𝐵𝐵 = 𝐶)
10074, 99syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐵, 𝐵} ∈ 𝐸) → 𝐵 = 𝐶)
101100expcom 414 . . . . . . . . . . . . . . . . . . 19 ({𝐵, 𝐵} ∈ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐵 = 𝐶))
102101adantl 482 . . . . . . . . . . . . . . . . . 18 (({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝐵} ∈ 𝐸) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐵 = 𝐶))
10372, 102sylbir 234 . . . . . . . . . . . . . . . . 17 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐵 = 𝐶))
104103com12 32 . . . . . . . . . . . . . . . 16 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸𝐵 = 𝐶))
1051043ad2ant3 1135 . . . . . . . . . . . . . . 15 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸𝐵 = 𝐶))
106105com12 32 . . . . . . . . . . . . . 14 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐶))
1071062a1i 12 . . . . . . . . . . . . 13 (𝑦 = 𝐶 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐶))))
108 eqeq2 2748 . . . . . . . . . . . . . . 15 (𝑦 = 𝐶 → (𝐵 = 𝑦𝐵 = 𝐶))
109108imbi2d 340 . . . . . . . . . . . . . 14 (𝑦 = 𝐶 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦) ↔ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐶)))
110109imbi2d 340 . . . . . . . . . . . . 13 (𝑦 = 𝐶 → (({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝐶))))
111107, 55, 1103imtr4d 293 . . . . . . . . . . . 12 (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦))))
11290, 97, 1113jaoi 1427 . . . . . . . . . . 11 ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦))))
113 preq1 4694 . . . . . . . . . . . . . . 15 (𝑥 = 𝐵 → {𝑥, 𝐴} = {𝐵, 𝐴})
114 preq1 4694 . . . . . . . . . . . . . . 15 (𝑥 = 𝐵 → {𝑥, 𝐵} = {𝐵, 𝐵})
115113, 114preq12d 4702 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
116115sseq1d 3975 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸))
117 eqeq1 2740 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝑥 = 𝑦𝐵 = 𝑦))
118117imbi2d 340 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦) ↔ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦)))
119116, 118imbi12d 344 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦))))
120119imbi2d 340 . . . . . . . . . . 11 (𝑥 = 𝐵 → (({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦))) ↔ ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐵 = 𝑦)))))
121112, 120syl5ibr 245 . . . . . . . . . 10 (𝑥 = 𝐵 → ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦)))))
12223pm2.24i 150 . . . . . . . . . . . . . . . . . . . . . 22 𝐴 = 𝐴𝐶 = 𝐴)
12322, 122sylbi 216 . . . . . . . . . . . . . . . . . . . . 21 (𝐴𝐴𝐶 = 𝐴)
12421, 123syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐶 = 𝐴)
125124expcom 414 . . . . . . . . . . . . . . . . . . 19 ({𝐴, 𝐴} ∈ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐶 = 𝐴))
126125adantr 481 . . . . . . . . . . . . . . . . . 18 (({𝐴, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝐵} ∈ 𝐸) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐶 = 𝐴))
12718, 126sylbir 234 . . . . . . . . . . . . . . . . 17 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → 𝐶 = 𝐴))
128127com12 32 . . . . . . . . . . . . . . . 16 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸𝐶 = 𝐴))
1291283ad2ant3 1135 . . . . . . . . . . . . . . 15 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸𝐶 = 𝐴))
130129com12 32 . . . . . . . . . . . . . 14 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐴))
131130a1i13 27 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐴))))
132 eqeq2 2748 . . . . . . . . . . . . . . 15 (𝑦 = 𝐴 → (𝐶 = 𝑦𝐶 = 𝐴))
133132imbi2d 340 . . . . . . . . . . . . . 14 (𝑦 = 𝐴 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦) ↔ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐴)))
134133imbi2d 340 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦)) ↔ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐴))))
135131, 11, 1343imtr4d 293 . . . . . . . . . . . 12 (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦))))
136 pm2.21 123 . . . . . . . . . . . . . . . . . . . . . . 23 𝐵 = 𝐵 → (𝐵 = 𝐵 → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 = 𝐵)))
13775, 136sylbi 216 . . . . . . . . . . . . . . . . . . . . . 22 (𝐵𝐵 → (𝐵 = 𝐵 → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 = 𝐵)))
13874, 76, 137mpisyl 21 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐵, 𝐵} ∈ 𝐸) → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 = 𝐵))
139138expcom 414 . . . . . . . . . . . . . . . . . . . 20 ({𝐵, 𝐵} ∈ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 = 𝐵)))
140139adantl 482 . . . . . . . . . . . . . . . . . . 19 (({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝐵} ∈ 𝐸) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 = 𝐵)))
14172, 140sylbir 234 . . . . . . . . . . . . . . . . . 18 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 = 𝐵)))
142141com13 88 . . . . . . . . . . . . . . . . 17 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸𝐶 = 𝐵)))
143142a1d 25 . . . . . . . . . . . . . . . 16 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((𝐴𝐵𝐴𝐶𝐵𝐶) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸𝐶 = 𝐵))))
1441433imp 1111 . . . . . . . . . . . . . . 15 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸𝐶 = 𝐵))
145144com12 32 . . . . . . . . . . . . . 14 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐵))
146145a1i13 27 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐵))))
147 eqeq2 2748 . . . . . . . . . . . . . . 15 (𝑦 = 𝐵 → (𝐶 = 𝑦𝐶 = 𝐵))
148147imbi2d 340 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦) ↔ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐵)))
149148imbi2d 340 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦)) ↔ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐵))))
150146, 37, 1493imtr4d 293 . . . . . . . . . . . 12 (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦))))
151 eqidd 2737 . . . . . . . . . . . . . . 15 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐶)
152151a1i 11 . . . . . . . . . . . . . 14 ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐶))
153152a1i13 27 . . . . . . . . . . . . 13 (𝑦 = 𝐶 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐶))))
154 eqeq2 2748 . . . . . . . . . . . . . . 15 (𝑦 = 𝐶 → (𝐶 = 𝑦𝐶 = 𝐶))
155154imbi2d 340 . . . . . . . . . . . . . 14 (𝑦 = 𝐶 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦) ↔ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐶)))
156155imbi2d 340 . . . . . . . . . . . . 13 (𝑦 = 𝐶 → (({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦)) ↔ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝐶))))
157153, 55, 1563imtr4d 293 . . . . . . . . . . . 12 (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦))))
158135, 150, 1573jaoi 1427 . . . . . . . . . . 11 ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦))))
159 preq1 4694 . . . . . . . . . . . . . . 15 (𝑥 = 𝐶 → {𝑥, 𝐴} = {𝐶, 𝐴})
160 preq1 4694 . . . . . . . . . . . . . . 15 (𝑥 = 𝐶 → {𝑥, 𝐵} = {𝐶, 𝐵})
161159, 160preq12d 4702 . . . . . . . . . . . . . 14 (𝑥 = 𝐶 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐶, 𝐴}, {𝐶, 𝐵}})
162161sseq1d 3975 . . . . . . . . . . . . 13 (𝑥 = 𝐶 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸))
163 eqeq1 2740 . . . . . . . . . . . . . 14 (𝑥 = 𝐶 → (𝑥 = 𝑦𝐶 = 𝑦))
164163imbi2d 340 . . . . . . . . . . . . 13 (𝑥 = 𝐶 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦) ↔ (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦)))
165162, 164imbi12d 344 . . . . . . . . . . . 12 (𝑥 = 𝐶 → (({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦)) ↔ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦))))
166165imbi2d 340 . . . . . . . . . . 11 (𝑥 = 𝐶 → (({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦))) ↔ ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝐶 = 𝑦)))))
167158, 166syl5ibr 245 . . . . . . . . . 10 (𝑥 = 𝐶 → ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦)))))
16869, 121, 1673jaoi 1427 . . . . . . . . 9 ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) → ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦)))))
169168com3l 89 . . . . . . . 8 ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦)))))
1704, 169sylbi 216 . . . . . . 7 (𝑦 ∈ {𝐴, 𝐵, 𝐶} → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸 → ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦)))))
171170imp 407 . . . . . 6 ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸) → ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦))))
172171com3l 89 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸) → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦))))
1732, 172sylbi 216 . . . 4 (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸) → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦))))
174173imp31 418 . . 3 (((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → 𝑥 = 𝑦))
175174com12 32 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → 𝑥 = 𝑦))
176175alrimivv 1931 1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ∀𝑥𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → 𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3o 1086  w3a 1087  wal 1539   = wceq 1541  wcel 2106  wne 2943  wss 3910  {cpr 4588  {ctp 4590  cfv 6496  Vtxcvtx 27947  Edgcedg 27998  USGraphcusgr 28100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-edg 27999  df-umgr 28034  df-usgr 28102
This theorem is referenced by:  frgr3vlem2  29218
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