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Theorem frgr3vlem2 27820
Description: Lemma 2 for frgr3v 27821. (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
frgr3vlem2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐸   𝑥,𝐺   𝑥,𝑉   𝑥,𝑋   𝑥,𝑌   𝑥,𝑍

Proof of Theorem frgr3vlem2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-reu 3089 . . 3 (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ∃!𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸))
2 eleq1w 2842 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑦 ∈ {𝐴, 𝐵, 𝐶}))
3 preq1 4539 . . . . . . . 8 (𝑥 = 𝑦 → {𝑥, 𝐴} = {𝑦, 𝐴})
4 preq1 4539 . . . . . . . 8 (𝑥 = 𝑦 → {𝑥, 𝐵} = {𝑦, 𝐵})
53, 4preq12d 4547 . . . . . . 7 (𝑥 = 𝑦 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝑦, 𝐴}, {𝑦, 𝐵}})
65sseq1d 3882 . . . . . 6 (𝑥 = 𝑦 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸))
72, 6anbi12d 621 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ↔ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)))
87eu4 2649 . . . 4 (∃!𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ↔ (∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ ∀𝑥𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → 𝑥 = 𝑦)))
9 frgr3v.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
10 frgr3v.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
119, 10frgr3vlem1 27819 . . . . . . 7 (((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))
12113expa 1098 . . . . . 6 ((((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))
1312biantrud 524 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ↔ (∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ ∀𝑥𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → 𝑥 = 𝑦))))
14 vex 3412 . . . . . . . . . . 11 𝑥 ∈ V
1514eltp 4496 . . . . . . . . . 10 (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶))
16 preq1 4539 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → {𝑥, 𝐴} = {𝐴, 𝐴})
17 preq1 4539 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
1816, 17preq12d 4547 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
1918sseq1d 3882 . . . . . . . . . . . 12 (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸))
20 prex 5185 . . . . . . . . . . . . . 14 {𝐴, 𝐴} ∈ V
21 prex 5185 . . . . . . . . . . . . . 14 {𝐴, 𝐵} ∈ V
2220, 21prss 4623 . . . . . . . . . . . . 13 (({𝐴, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝐵} ∈ 𝐸) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸)
2310usgredgne 26703 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ USGraph ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴𝐴)
2423adantll 701 . . . . . . . . . . . . . . . . . 18 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴𝐴)
25 df-ne 2962 . . . . . . . . . . . . . . . . . . 19 (𝐴𝐴 ↔ ¬ 𝐴 = 𝐴)
26 eqid 2772 . . . . . . . . . . . . . . . . . . . 20 𝐴 = 𝐴
2726pm2.24i 148 . . . . . . . . . . . . . . . . . . 19 𝐴 = 𝐴 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
2825, 27sylbi 209 . . . . . . . . . . . . . . . . . 18 (𝐴𝐴 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
2924, 28syl 17 . . . . . . . . . . . . . . . . 17 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
3029ex 405 . . . . . . . . . . . . . . . 16 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({𝐴, 𝐴} ∈ 𝐸 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
3130adantl 474 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐴, 𝐴} ∈ 𝐸 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
3231com12 32 . . . . . . . . . . . . . 14 ({𝐴, 𝐴} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
3332adantr 473 . . . . . . . . . . . . 13 (({𝐴, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝐵} ∈ 𝐸) → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
3422, 33sylbir 227 . . . . . . . . . . . 12 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
3519, 34syl6bi 245 . . . . . . . . . . 11 (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
36 preq1 4539 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → {𝑥, 𝐴} = {𝐵, 𝐴})
37 preq1 4539 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → {𝑥, 𝐵} = {𝐵, 𝐵})
3836, 37preq12d 4547 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
3938sseq1d 3882 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸))
40 prex 5185 . . . . . . . . . . . . . 14 {𝐵, 𝐴} ∈ V
41 prex 5185 . . . . . . . . . . . . . 14 {𝐵, 𝐵} ∈ V
4240, 41prss 4623 . . . . . . . . . . . . 13 (({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝐵} ∈ 𝐸) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸)
4310usgredgne 26703 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ USGraph ∧ {𝐵, 𝐵} ∈ 𝐸) → 𝐵𝐵)
4443adantll 701 . . . . . . . . . . . . . . . . . 18 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐵, 𝐵} ∈ 𝐸) → 𝐵𝐵)
45 df-ne 2962 . . . . . . . . . . . . . . . . . . 19 (𝐵𝐵 ↔ ¬ 𝐵 = 𝐵)
46 eqid 2772 . . . . . . . . . . . . . . . . . . . 20 𝐵 = 𝐵
4746pm2.24i 148 . . . . . . . . . . . . . . . . . . 19 𝐵 = 𝐵 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
4845, 47sylbi 209 . . . . . . . . . . . . . . . . . 18 (𝐵𝐵 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
4944, 48syl 17 . . . . . . . . . . . . . . . . 17 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐵, 𝐵} ∈ 𝐸) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
5049ex 405 . . . . . . . . . . . . . . . 16 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({𝐵, 𝐵} ∈ 𝐸 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
5150adantl 474 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐵, 𝐵} ∈ 𝐸 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
5251com12 32 . . . . . . . . . . . . . 14 ({𝐵, 𝐵} ∈ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
5352adantl 474 . . . . . . . . . . . . 13 (({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝐵} ∈ 𝐸) → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
5442, 53sylbir 227 . . . . . . . . . . . 12 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
5539, 54syl6bi 245 . . . . . . . . . . 11 (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
56 preq1 4539 . . . . . . . . . . . . . 14 (𝑥 = 𝐶 → {𝑥, 𝐴} = {𝐶, 𝐴})
57 preq1 4539 . . . . . . . . . . . . . 14 (𝑥 = 𝐶 → {𝑥, 𝐵} = {𝐶, 𝐵})
5856, 57preq12d 4547 . . . . . . . . . . . . 13 (𝑥 = 𝐶 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐶, 𝐴}, {𝐶, 𝐵}})
5958sseq1d 3882 . . . . . . . . . . . 12 (𝑥 = 𝐶 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸))
60 prex 5185 . . . . . . . . . . . . . 14 {𝐶, 𝐴} ∈ V
61 prex 5185 . . . . . . . . . . . . . 14 {𝐶, 𝐵} ∈ V
6260, 61prss 4623 . . . . . . . . . . . . 13 (({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸) ↔ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸)
63 ax-1 6 . . . . . . . . . . . . 13 (({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸) → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
6462, 63sylbir 227 . . . . . . . . . . . 12 ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
6559, 64syl6bi 245 . . . . . . . . . . 11 (𝑥 = 𝐶 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
6635, 55, 653jaoi 1407 . . . . . . . . . 10 ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
6715, 66sylbi 209 . . . . . . . . 9 (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
6867imp 398 . . . . . . . 8 ((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
6968com12 32 . . . . . . 7 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
7069exlimdv 1892 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
71 prssi 4624 . . . . . . . . . . 11 (({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸) → {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸)
7271adantl 474 . . . . . . . . . 10 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)) → {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸)
73723mix3d 1318 . . . . . . . . 9 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)) → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 ∨ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸))
7419, 39, 59rextpg 4505 . . . . . . . . . 10 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 ∨ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸)))
7574ad3antrrr 717 . . . . . . . . 9 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 ∨ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸)))
7673, 75mpbird 249 . . . . . . . 8 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)) → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸)
77 df-rex 3088 . . . . . . . 8 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸))
7876, 77sylib 210 . . . . . . 7 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)) → ∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸))
7978ex 405 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸) → ∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸)))
8070, 79impbid 204 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
8113, 80bitr3d 273 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ((∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ ∀𝑥𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)) → 𝑥 = 𝑦)) ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
828, 81syl5bb 275 . . 3 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (∃!𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
831, 82syl5bb 275 . 2 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
8483ex 405 1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  w3o 1067  w3a 1068  wal 1505   = wceq 1507  wex 1742  wcel 2050  ∃!weu 2583  wne 2961  wrex 3083  ∃!wreu 3084  wss 3823  {cpr 4437  {ctp 4439  cfv 6185  Vtxcvtx 26496  Edgcedg 26547  USGraphcusgr 26649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-oadd 7907  df-er 8087  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-dju 9122  df-card 9160  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-nn 11438  df-2 11501  df-n0 11706  df-z 11792  df-uz 12057  df-fz 12707  df-hash 13504  df-edg 26548  df-umgr 26583  df-usgr 26651
This theorem is referenced by:  frgr3v  27821
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