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Theorem frgra3vlem2 26528
 Description: Lemma 2 for frgra3v 26529. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
Assertion
Ref Expression
frgra3vlem2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐸   𝑥,𝑋   𝑥,𝑌   𝑥,𝑍

Proof of Theorem frgra3vlem2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-reu 2903 . . 3 (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ∃!𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
2 eleq1 2676 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ 𝑦 ∈ {𝐴, 𝐵, 𝐶}))
3 preq1 4212 . . . . . . . 8 (𝑥 = 𝑦 → {𝑥, 𝐴} = {𝑦, 𝐴})
4 preq1 4212 . . . . . . . 8 (𝑥 = 𝑦 → {𝑥, 𝐵} = {𝑦, 𝐵})
53, 4preq12d 4220 . . . . . . 7 (𝑥 = 𝑦 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝑦, 𝐴}, {𝑦, 𝐵}})
65sseq1d 3595 . . . . . 6 (𝑥 = 𝑦 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸))
72, 6anbi12d 743 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ↔ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸)))
87eu4 2506 . . . 4 (∃!𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ↔ (∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ∧ ∀𝑥𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸)) → 𝑥 = 𝑦)))
9 frgra3vlem1 26527 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ∀𝑥𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸)) → 𝑥 = 𝑦))
109biantrud 527 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ↔ (∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ∧ ∀𝑥𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸)) → 𝑥 = 𝑦))))
11 vex 3176 . . . . . . . . . . 11 𝑥 ∈ V
1211eltp 4177 . . . . . . . . . 10 (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶))
13 preq1 4212 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → {𝑥, 𝐴} = {𝐴, 𝐴})
14 preq1 4212 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
1513, 14preq12d 4220 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
1615sseq1d 3595 . . . . . . . . . . . 12 (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸))
17 prex 4836 . . . . . . . . . . . . . 14 {𝐴, 𝐴} ∈ V
18 prex 4836 . . . . . . . . . . . . . 14 {𝐴, 𝐵} ∈ V
1917, 18prss 4291 . . . . . . . . . . . . 13 (({𝐴, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸)
20 usgraedgrn 25910 . . . . . . . . . . . . . . . . . 18 (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐴, 𝐴} ∈ ran 𝐸) → 𝐴𝐴)
21 df-ne 2782 . . . . . . . . . . . . . . . . . . 19 (𝐴𝐴 ↔ ¬ 𝐴 = 𝐴)
22 eqid 2610 . . . . . . . . . . . . . . . . . . . 20 𝐴 = 𝐴
2322pm2.24i 145 . . . . . . . . . . . . . . . . . . 19 𝐴 = 𝐴 → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))
2421, 23sylbi 206 . . . . . . . . . . . . . . . . . 18 (𝐴𝐴 → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))
2520, 24syl 17 . . . . . . . . . . . . . . . . 17 (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐴, 𝐴} ∈ ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))
2625ex 449 . . . . . . . . . . . . . . . 16 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({𝐴, 𝐴} ∈ ran 𝐸 → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
2726adantl 481 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐴, 𝐴} ∈ ran 𝐸 → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
2827com12 32 . . . . . . . . . . . . . 14 ({𝐴, 𝐴} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
2928adantr 480 . . . . . . . . . . . . 13 (({𝐴, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
3019, 29sylbir 224 . . . . . . . . . . . 12 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
3116, 30syl6bi 242 . . . . . . . . . . 11 (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))))
32 preq1 4212 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → {𝑥, 𝐴} = {𝐵, 𝐴})
33 preq1 4212 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → {𝑥, 𝐵} = {𝐵, 𝐵})
3432, 33preq12d 4220 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
3534sseq1d 3595 . . . . . . . . . . . 12 (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸))
36 prex 4836 . . . . . . . . . . . . . 14 {𝐵, 𝐴} ∈ V
37 prex 4836 . . . . . . . . . . . . . 14 {𝐵, 𝐵} ∈ V
3836, 37prss 4291 . . . . . . . . . . . . 13 (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸)
39 usgraedgrn 25910 . . . . . . . . . . . . . . . . . 18 (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → 𝐵𝐵)
40 df-ne 2782 . . . . . . . . . . . . . . . . . . 19 (𝐵𝐵 ↔ ¬ 𝐵 = 𝐵)
41 eqid 2610 . . . . . . . . . . . . . . . . . . . 20 𝐵 = 𝐵
4241pm2.24i 145 . . . . . . . . . . . . . . . . . . 19 𝐵 = 𝐵 → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))
4340, 42sylbi 206 . . . . . . . . . . . . . . . . . 18 (𝐵𝐵 → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))
4439, 43syl 17 . . . . . . . . . . . . . . . . 17 (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))
4544ex 449 . . . . . . . . . . . . . . . 16 ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({𝐵, 𝐵} ∈ ran 𝐸 → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
4645adantl 481 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐵, 𝐵} ∈ ran 𝐸 → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
4746com12 32 . . . . . . . . . . . . . 14 ({𝐵, 𝐵} ∈ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
4847adantl 481 . . . . . . . . . . . . 13 (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
4938, 48sylbir 224 . . . . . . . . . . . 12 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
5035, 49syl6bi 242 . . . . . . . . . . 11 (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))))
51 preq1 4212 . . . . . . . . . . . . . 14 (𝑥 = 𝐶 → {𝑥, 𝐴} = {𝐶, 𝐴})
52 preq1 4212 . . . . . . . . . . . . . 14 (𝑥 = 𝐶 → {𝑥, 𝐵} = {𝐶, 𝐵})
5351, 52preq12d 4220 . . . . . . . . . . . . 13 (𝑥 = 𝐶 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐶, 𝐴}, {𝐶, 𝐵}})
5453sseq1d 3595 . . . . . . . . . . . 12 (𝑥 = 𝐶 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸))
55 prex 4836 . . . . . . . . . . . . . 14 {𝐶, 𝐴} ∈ V
56 prex 4836 . . . . . . . . . . . . . 14 {𝐶, 𝐵} ∈ V
5755, 56prss 4291 . . . . . . . . . . . . 13 (({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ↔ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸)
58 ax-1 6 . . . . . . . . . . . . 13 (({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
5957, 58sylbir 224 . . . . . . . . . . . 12 ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
6054, 59syl6bi 242 . . . . . . . . . . 11 (𝑥 = 𝐶 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))))
6131, 50, 603jaoi 1383 . . . . . . . . . 10 ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))))
6212, 61sylbi 206 . . . . . . . . 9 (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))))
6362imp 444 . . . . . . . 8 ((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
6463com12 32 . . . . . . 7 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
6564exlimdv 1848 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) → ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
66 prssi 4293 . . . . . . . . . . 11 (({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) → {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸)
6766adantl 481 . . . . . . . . . 10 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)) → {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸)
68673mix3d 1231 . . . . . . . . 9 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)) → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 ∨ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸))
6916, 35, 54rextpg 4184 . . . . . . . . . 10 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 ∨ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸)))
7069ad3antrrr 762 . . . . . . . . 9 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)) → (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 ∨ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸)))
7168, 70mpbird 246 . . . . . . . 8 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)) → ∃𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸)
72 df-rex 2902 . . . . . . . 8 (∃𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
7371, 72sylib 207 . . . . . . 7 (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)) → ∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
7473ex 449 . . . . . 6 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) → ∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸)))
7565, 74impbid 201 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ↔ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
7610, 75bitr3d 269 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ((∃𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ∧ ∀𝑥𝑦(((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸)) → 𝑥 = 𝑦)) ↔ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
778, 76syl5bb 271 . . 3 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∃!𝑥(𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ↔ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
781, 77syl5bb 271 . 2 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
7978ex 449 1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → (∃!𝑥 ∈ {𝐴, 𝐵, 𝐶} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458   ≠ wne 2780  ∃wrex 2897  ∃!wreu 2898   ⊆ wss 3540  {cpr 4127  {ctp 4129   class class class wbr 4583  ran crn 5039   USGrph cusg 25859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862 This theorem is referenced by:  frgra3v  26529
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