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Theorem frgr3vlem2 29387
Description: Lemma 2 for frgr3v 29388. (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 3376 . . 3 (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ∃!𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸))
2 eleq1w 2815 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑦 ∈ {𝐴, 𝐵, 𝐶}))
3 preq1 4727 . . . . . . . 8 (𝑥 = 𝑦 → {𝑥, 𝐴} = {𝑦, 𝐴})
4 preq1 4727 . . . . . . . 8 (𝑥 = 𝑦 → {𝑥, 𝐵} = {𝑦, 𝐵})
53, 4preq12d 4735 . . . . . . 7 (𝑥 = 𝑦 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝑦, 𝐴}, {𝑦, 𝐵}})
65sseq1d 4006 . . . . . 6 (𝑥 = 𝑦 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸))
72, 6anbi12d 631 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸) ↔ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ 𝐸)))
87eu4 2610 . . . 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 29386 . . . . . . 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 1118 . . . . . 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 532 . . . . 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 3474 . . . . . . . . . . 11 𝑥 ∈ V
1514eltp 4682 . . . . . . . . . 10 (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶))
16 preq1 4727 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → {𝑥, 𝐴} = {𝐴, 𝐴})
17 preq1 4727 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
1816, 17preq12d 4735 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
1918sseq1d 4006 . . . . . . . . . . . 12 (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸))
20 prex 5422 . . . . . . . . . . . . . 14 {𝐴, 𝐴} ∈ V
21 prex 5422 . . . . . . . . . . . . . 14 {𝐴, 𝐵} ∈ V
2220, 21prss 4813 . . . . . . . . . . . . 13 (({𝐴, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝐵} ∈ 𝐸) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸)
2310usgredgne 28323 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ USGraph ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴𝐴)
2423adantll 712 . . . . . . . . . . . . . . . . . 18 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐴, 𝐴} ∈ 𝐸) → 𝐴𝐴)
25 df-ne 2940 . . . . . . . . . . . . . . . . . . 19 (𝐴𝐴 ↔ ¬ 𝐴 = 𝐴)
26 eqid 2731 . . . . . . . . . . . . . . . . . . . 20 𝐴 = 𝐴
2726pm2.24i 150 . . . . . . . . . . . . . . . . . . 19 𝐴 = 𝐴 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
2825, 27sylbi 216 . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . 16 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({𝐴, 𝐴} ∈ 𝐸 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
3130adantl 482 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 13 (({𝐴, 𝐴} ∈ 𝐸 ∧ {𝐴, 𝐵} ∈ 𝐸) → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
3422, 33sylbir 234 . . . . . . . . . . . 12 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
3519, 34syl6bi 252 . . . . . . . . . . 11 (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
36 preq1 4727 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → {𝑥, 𝐴} = {𝐵, 𝐴})
37 preq1 4727 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → {𝑥, 𝐵} = {𝐵, 𝐵})
3836, 37preq12d 4735 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
3938sseq1d 4006 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸))
40 prex 5422 . . . . . . . . . . . . . 14 {𝐵, 𝐴} ∈ V
41 prex 5422 . . . . . . . . . . . . . 14 {𝐵, 𝐵} ∈ V
4240, 41prss 4813 . . . . . . . . . . . . 13 (({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝐵} ∈ 𝐸) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸)
4310usgredgne 28323 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ USGraph ∧ {𝐵, 𝐵} ∈ 𝐸) → 𝐵𝐵)
4443adantll 712 . . . . . . . . . . . . . . . . . 18 (((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) ∧ {𝐵, 𝐵} ∈ 𝐸) → 𝐵𝐵)
45 df-ne 2940 . . . . . . . . . . . . . . . . . . 19 (𝐵𝐵 ↔ ¬ 𝐵 = 𝐵)
46 eqid 2731 . . . . . . . . . . . . . . . . . . . 20 𝐵 = 𝐵
4746pm2.24i 150 . . . . . . . . . . . . . . . . . . 19 𝐵 = 𝐵 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))
4845, 47sylbi 216 . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . 16 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph) → ({𝐵, 𝐵} ∈ 𝐸 → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
5150adantl 482 . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . 13 (({𝐵, 𝐴} ∈ 𝐸 ∧ {𝐵, 𝐵} ∈ 𝐸) → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
5442, 53sylbir 234 . . . . . . . . . . . 12 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
5539, 54syl6bi 252 . . . . . . . . . . 11 (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸))))
56 preq1 4727 . . . . . . . . . . . . . 14 (𝑥 = 𝐶 → {𝑥, 𝐴} = {𝐶, 𝐴})
57 preq1 4727 . . . . . . . . . . . . . 14 (𝑥 = 𝐶 → {𝑥, 𝐵} = {𝐶, 𝐵})
5856, 57preq12d 4735 . . . . . . . . . . . . 13 (𝑥 = 𝐶 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐶, 𝐴}, {𝐶, 𝐵}})
5958sseq1d 4006 . . . . . . . . . . . 12 (𝑥 = 𝐶 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸))
60 prex 5422 . . . . . . . . . . . . . 14 {𝐶, 𝐴} ∈ V
61 prex 5422 . . . . . . . . . . . . . 14 {𝐶, 𝐵} ∈ V
6260, 61prss 4813 . . . . . . . . . . . . 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 234 . . . . . . . . . . . 12 ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) → ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)))
6559, 64syl6bi 252 . . . . . . . . . . 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 1427 . . . . . . . . . 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 216 . . . . . . . . 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 407 . . . . . . . 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 1936 . . . . . 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 4814 . . . . . . . . . . 11 (({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸) → {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸)
7271adantl 482 . . . . . . . . . 10 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph)) ∧ ({𝐶, 𝐴} ∈ 𝐸 ∧ {𝐶, 𝐵} ∈ 𝐸)) → {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ 𝐸)
73723mix3d 1338 . . . . . . . . 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 4693 . . . . . . . . . 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 728 . . . . . . . . 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 256 . . . . . . . 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 3070 . . . . . . . 8 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸 ↔ ∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ 𝐸))
7876, 77sylib 217 . . . . . . 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 413 . . . . . 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 211 . . . . 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 280 . . . 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, 81bitrid 282 . . 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, 82bitrid 282 . 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 413 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 205  wa 396  w3o 1086  w3a 1087  wal 1539   = wceq 1541  wex 1781  wcel 2106  ∃!weu 2561  wne 2939  wrex 3069  ∃!wreu 3373  wss 3941  {cpr 4621  {ctp 4623  cfv 6529  Vtxcvtx 28116  Edgcedg 28167  USGraphcusgr 28269
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 2702  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7705  ax-cnex 11145  ax-resscn 11146  ax-1cn 11147  ax-icn 11148  ax-addcl 11149  ax-addrcl 11150  ax-mulcl 11151  ax-mulrcl 11152  ax-mulcom 11153  ax-addass 11154  ax-mulass 11155  ax-distr 11156  ax-i2m1 11157  ax-1ne0 11158  ax-1rid 11159  ax-rnegex 11160  ax-rrecex 11161  ax-cnre 11162  ax-pre-lttri 11163  ax-pre-lttrn 11164  ax-pre-ltadd 11165  ax-pre-mulgt0 11166
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3430  df-v 3472  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4520  df-pw 4595  df-sn 4620  df-pr 4622  df-tp 4624  df-op 4626  df-uni 4899  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6286  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6531  df-fn 6532  df-f 6533  df-f1 6534  df-fo 6535  df-f1o 6536  df-fv 6537  df-riota 7346  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7836  df-1st 7954  df-2nd 7955  df-frecs 8245  df-wrecs 8276  df-recs 8350  df-rdg 8389  df-1o 8445  df-oadd 8449  df-er 8683  df-en 8920  df-dom 8921  df-sdom 8922  df-fin 8923  df-dju 9875  df-card 9913  df-pnf 11229  df-mnf 11230  df-xr 11231  df-ltxr 11232  df-le 11233  df-sub 11425  df-neg 11426  df-nn 12192  df-2 12254  df-n0 12452  df-z 12538  df-uz 12802  df-fz 13464  df-hash 14270  df-edg 28168  df-umgr 28203  df-usgr 28271
This theorem is referenced by:  frgr3v  29388
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