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Theorem frgra3vlem1 26527
 Description: Lemma 1 for frgra3v 26529. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
Assertion
Ref Expression
frgra3vlem1 ((((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))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐸,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝑍,𝑦

Proof of Theorem frgra3vlem1
StepHypRef Expression
1 vex 3176 . . . . . 6 𝑥 ∈ V
21eltp 4177 . . . . 5 (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶))
3 vex 3176 . . . . . . . . 9 𝑦 ∈ V
43eltp 4177 . . . . . . . 8 (𝑦 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶))
5 eqidd 2611 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐴)
65a1i 11 . . . . . . . . . . . . . 14 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐴))
762a1i 12 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐴))))
8 preq1 4212 . . . . . . . . . . . . . . 15 (𝑦 = 𝐴 → {𝑦, 𝐴} = {𝐴, 𝐴})
9 preq1 4212 . . . . . . . . . . . . . . 15 (𝑦 = 𝐴 → {𝑦, 𝐵} = {𝐴, 𝐵})
108, 9preq12d 4220 . . . . . . . . . . . . . 14 (𝑦 = 𝐴 → {{𝑦, 𝐴}, {𝑦, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
1110sseq1d 3595 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸))
12 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 𝐴 → (𝐴 = 𝑦𝐴 = 𝐴))
1312imbi2d 329 . . . . . . . . . . . . . 14 (𝑦 = 𝐴 → (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐴)))
1413imbi2d 329 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐴))))
157, 11, 143imtr4d 282 . . . . . . . . . . . 12 (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦))))
16 prex 4836 . . . . . . . . . . . . . . . . 17 {𝐴, 𝐴} ∈ V
17 prex 4836 . . . . . . . . . . . . . . . . 17 {𝐴, 𝐵} ∈ V
1816, 17prss 4291 . . . . . . . . . . . . . . . 16 (({𝐴, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸)
19 usgraedgrn 25910 . . . . . . . . . . . . . . . . . . 19 (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐴, 𝐴} ∈ ran 𝐸) → 𝐴𝐴)
20 df-ne 2782 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝐴 ↔ ¬ 𝐴 = 𝐴)
21 eqid 2610 . . . . . . . . . . . . . . . . . . . . 21 𝐴 = 𝐴
2221pm2.24i 145 . . . . . . . . . . . . . . . . . . . 20 𝐴 = 𝐴𝐴 = 𝐵)
2320, 22sylbi 206 . . . . . . . . . . . . . . . . . . 19 (𝐴𝐴𝐴 = 𝐵)
2419, 23syl 17 . . . . . . . . . . . . . . . . . 18 (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐴, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝐵)
2524expcom 450 . . . . . . . . . . . . . . . . 17 ({𝐴, 𝐴} ∈ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸𝐴 = 𝐵))
2625adantr 480 . . . . . . . . . . . . . . . 16 (({𝐴, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸𝐴 = 𝐵))
2718, 26sylbir 224 . . . . . . . . . . . . . . 15 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸𝐴 = 𝐵))
2827adantld 482 . . . . . . . . . . . . . 14 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐵))
29282a1i 12 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐵))))
30 preq1 4212 . . . . . . . . . . . . . . 15 (𝑦 = 𝐵 → {𝑦, 𝐴} = {𝐵, 𝐴})
31 preq1 4212 . . . . . . . . . . . . . . 15 (𝑦 = 𝐵 → {𝑦, 𝐵} = {𝐵, 𝐵})
3230, 31preq12d 4220 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → {{𝑦, 𝐴}, {𝑦, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
3332sseq1d 3595 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸))
34 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
3534imbi2d 329 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐵)))
3635imbi2d 329 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐵))))
3729, 33, 363imtr4d 282 . . . . . . . . . . . 12 (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦))))
3821pm2.24i 145 . . . . . . . . . . . . . . . . . . . 20 𝐴 = 𝐴𝐴 = 𝐶)
3920, 38sylbi 206 . . . . . . . . . . . . . . . . . . 19 (𝐴𝐴𝐴 = 𝐶)
4019, 39syl 17 . . . . . . . . . . . . . . . . . 18 (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐴, 𝐴} ∈ ran 𝐸) → 𝐴 = 𝐶)
4140expcom 450 . . . . . . . . . . . . . . . . 17 ({𝐴, 𝐴} ∈ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸𝐴 = 𝐶))
4241adantr 480 . . . . . . . . . . . . . . . 16 (({𝐴, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸𝐴 = 𝐶))
4318, 42sylbir 224 . . . . . . . . . . . . . . 15 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸𝐴 = 𝐶))
4443adantld 482 . . . . . . . . . . . . . 14 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐶))
45442a1i 12 . . . . . . . . . . . . 13 (𝑦 = 𝐶 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐶))))
46 preq1 4212 . . . . . . . . . . . . . . 15 (𝑦 = 𝐶 → {𝑦, 𝐴} = {𝐶, 𝐴})
47 preq1 4212 . . . . . . . . . . . . . . 15 (𝑦 = 𝐶 → {𝑦, 𝐵} = {𝐶, 𝐵})
4846, 47preq12d 4220 . . . . . . . . . . . . . 14 (𝑦 = 𝐶 → {{𝑦, 𝐴}, {𝑦, 𝐵}} = {{𝐶, 𝐴}, {𝐶, 𝐵}})
4948sseq1d 3595 . . . . . . . . . . . . 13 (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸))
50 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 𝐶 → (𝐴 = 𝑦𝐴 = 𝐶))
5150imbi2d 329 . . . . . . . . . . . . . 14 (𝑦 = 𝐶 → (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐶)))
5251imbi2d 329 . . . . . . . . . . . . 13 (𝑦 = 𝐶 → (({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝐶))))
5345, 49, 523imtr4d 282 . . . . . . . . . . . 12 (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦))))
5415, 37, 533jaoi 1383 . . . . . . . . . . 11 ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦))))
55 preq1 4212 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → {𝑥, 𝐴} = {𝐴, 𝐴})
56 preq1 4212 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
5755, 56preq12d 4220 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
5857sseq1d 3595 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸))
59 eqeq1 2614 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
6059imbi2d 329 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦)))
6158, 60imbi12d 333 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦))))
6261imbi2d 329 . . . . . . . . . . 11 (𝑥 = 𝐴 → (({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦))) ↔ ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐴 = 𝑦)))))
6354, 62syl5ibr 235 . . . . . . . . . 10 (𝑥 = 𝐴 → ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)))))
64 prex 4836 . . . . . . . . . . . . . . . . 17 {𝐵, 𝐴} ∈ V
65 prex 4836 . . . . . . . . . . . . . . . . 17 {𝐵, 𝐵} ∈ V
6664, 65prss 4291 . . . . . . . . . . . . . . . 16 (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸)
67 usgraedgrn 25910 . . . . . . . . . . . . . . . . . . 19 (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → 𝐵𝐵)
68 df-ne 2782 . . . . . . . . . . . . . . . . . . . 20 (𝐵𝐵 ↔ ¬ 𝐵 = 𝐵)
69 eqid 2610 . . . . . . . . . . . . . . . . . . . . 21 𝐵 = 𝐵
7069pm2.24i 145 . . . . . . . . . . . . . . . . . . . 20 𝐵 = 𝐵𝐵 = 𝐴)
7168, 70sylbi 206 . . . . . . . . . . . . . . . . . . 19 (𝐵𝐵𝐵 = 𝐴)
7267, 71syl 17 . . . . . . . . . . . . . . . . . 18 (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝐴)
7372expcom 450 . . . . . . . . . . . . . . . . 17 ({𝐵, 𝐵} ∈ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸𝐵 = 𝐴))
7473adantl 481 . . . . . . . . . . . . . . . 16 (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸𝐵 = 𝐴))
7566, 74sylbir 224 . . . . . . . . . . . . . . 15 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸𝐵 = 𝐴))
7675adantld 482 . . . . . . . . . . . . . 14 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐴))
77762a1i 12 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐴))))
78 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 𝐴 → (𝐵 = 𝑦𝐵 = 𝐴))
7978imbi2d 329 . . . . . . . . . . . . . 14 (𝑦 = 𝐴 → (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐴)))
8079imbi2d 329 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐴))))
8177, 11, 803imtr4d 282 . . . . . . . . . . . 12 (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦))))
82 eqidd 2611 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐵)
8382a1i 11 . . . . . . . . . . . . . 14 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐵))
84832a1i 12 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐵))))
85 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 𝐵 → (𝐵 = 𝑦𝐵 = 𝐵))
8685imbi2d 329 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐵)))
8786imbi2d 329 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐵))))
8884, 33, 873imtr4d 282 . . . . . . . . . . . 12 (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦))))
8969pm2.24i 145 . . . . . . . . . . . . . . . . . . . 20 𝐵 = 𝐵𝐵 = 𝐶)
9068, 89sylbi 206 . . . . . . . . . . . . . . . . . . 19 (𝐵𝐵𝐵 = 𝐶)
9167, 90syl 17 . . . . . . . . . . . . . . . . . 18 (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → 𝐵 = 𝐶)
9291expcom 450 . . . . . . . . . . . . . . . . 17 ({𝐵, 𝐵} ∈ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸𝐵 = 𝐶))
9392adantl 481 . . . . . . . . . . . . . . . 16 (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸𝐵 = 𝐶))
9466, 93sylbir 224 . . . . . . . . . . . . . . 15 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸𝐵 = 𝐶))
9594adantld 482 . . . . . . . . . . . . . 14 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐶))
96952a1i 12 . . . . . . . . . . . . 13 (𝑦 = 𝐶 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐶))))
97 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 𝐶 → (𝐵 = 𝑦𝐵 = 𝐶))
9897imbi2d 329 . . . . . . . . . . . . . 14 (𝑦 = 𝐶 → (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐶)))
9998imbi2d 329 . . . . . . . . . . . . 13 (𝑦 = 𝐶 → (({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝐶))))
10096, 49, 993imtr4d 282 . . . . . . . . . . . 12 (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦))))
10181, 88, 1003jaoi 1383 . . . . . . . . . . 11 ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦))))
102 preq1 4212 . . . . . . . . . . . . . . 15 (𝑥 = 𝐵 → {𝑥, 𝐴} = {𝐵, 𝐴})
103 preq1 4212 . . . . . . . . . . . . . . 15 (𝑥 = 𝐵 → {𝑥, 𝐵} = {𝐵, 𝐵})
104102, 103preq12d 4220 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
105104sseq1d 3595 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸))
106 eqeq1 2614 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝑥 = 𝑦𝐵 = 𝑦))
107106imbi2d 329 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦)))
108105, 107imbi12d 333 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)) ↔ ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦))))
109108imbi2d 329 . . . . . . . . . . 11 (𝑥 = 𝐵 → (({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦))) ↔ ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐵 = 𝑦)))))
110101, 109syl5ibr 235 . . . . . . . . . 10 (𝑥 = 𝐵 → ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)))))
11121pm2.24i 145 . . . . . . . . . . . . . . . . . . . . 21 𝐴 = 𝐴𝐶 = 𝐴)
11220, 111sylbi 206 . . . . . . . . . . . . . . . . . . . 20 (𝐴𝐴𝐶 = 𝐴)
11319, 112syl 17 . . . . . . . . . . . . . . . . . . 19 (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐴, 𝐴} ∈ ran 𝐸) → 𝐶 = 𝐴)
114113expcom 450 . . . . . . . . . . . . . . . . . 18 ({𝐴, 𝐴} ∈ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸𝐶 = 𝐴))
115114adantr 480 . . . . . . . . . . . . . . . . 17 (({𝐴, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸𝐶 = 𝐴))
11618, 115sylbir 224 . . . . . . . . . . . . . . . 16 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸𝐶 = 𝐴))
117116adantld 482 . . . . . . . . . . . . . . 15 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐴))
118117a1d 25 . . . . . . . . . . . . . 14 ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐴)))
119118a1i 11 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐴))))
120 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 𝐴 → (𝐶 = 𝑦𝐶 = 𝐴))
121120imbi2d 329 . . . . . . . . . . . . . 14 (𝑦 = 𝐴 → (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐴)))
122121imbi2d 329 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦)) ↔ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐴))))
123119, 11, 1223imtr4d 282 . . . . . . . . . . . 12 (𝑦 = 𝐴 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦))))
124 pm2.21 119 . . . . . . . . . . . . . . . . . . . . . . . 24 𝐵 = 𝐵 → (𝐵 = 𝐵 → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 = 𝐵)))
12568, 124sylbi 206 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐵𝐵 → (𝐵 = 𝐵 → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 = 𝐵)))
12667, 69, 125mpisyl 21 . . . . . . . . . . . . . . . . . . . . . 22 (({𝐴, 𝐵, 𝐶} USGrph 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 = 𝐵))
127126expcom 450 . . . . . . . . . . . . . . . . . . . . 21 ({𝐵, 𝐵} ∈ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 = 𝐵)))
128127adantl 481 . . . . . . . . . . . . . . . . . . . 20 (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 = 𝐵)))
12966, 128sylbir 224 . . . . . . . . . . . . . . . . . . 19 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶 = 𝐵)))
130129com13 86 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸𝐶 = 𝐵)))
131130adantr 480 . . . . . . . . . . . . . . . . 17 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} USGrph 𝐸 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸𝐶 = 𝐵)))
132131imp 444 . . . . . . . . . . . . . . . 16 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸𝐶 = 𝐵))
133132com12 32 . . . . . . . . . . . . . . 15 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐵))
134133a1d 25 . . . . . . . . . . . . . 14 ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐵)))
135134a1i 11 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → ({{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐵))))
136 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 𝐵 → (𝐶 = 𝑦𝐶 = 𝐵))
137136imbi2d 329 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐵)))
138137imbi2d 329 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦)) ↔ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐵))))
139135, 33, 1383imtr4d 282 . . . . . . . . . . . 12 (𝑦 = 𝐵 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦))))
140 eqidd 2611 . . . . . . . . . . . . . . 15 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐶)
141140a1i 11 . . . . . . . . . . . . . 14 ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐶))
1421412a1i 12 . . . . . . . . . . . . 13 (𝑦 = 𝐶 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐶))))
143 eqeq2 2621 . . . . . . . . . . . . . . 15 (𝑦 = 𝐶 → (𝐶 = 𝑦𝐶 = 𝐶))
144143imbi2d 329 . . . . . . . . . . . . . 14 (𝑦 = 𝐶 → (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐶)))
145144imbi2d 329 . . . . . . . . . . . . 13 (𝑦 = 𝐶 → (({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦)) ↔ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝐶))))
146142, 49, 1453imtr4d 282 . . . . . . . . . . . 12 (𝑦 = 𝐶 → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦))))
147123, 139, 1463jaoi 1383 . . . . . . . . . . 11 ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦))))
148 preq1 4212 . . . . . . . . . . . . . . 15 (𝑥 = 𝐶 → {𝑥, 𝐴} = {𝐶, 𝐴})
149 preq1 4212 . . . . . . . . . . . . . . 15 (𝑥 = 𝐶 → {𝑥, 𝐵} = {𝐶, 𝐵})
150148, 149preq12d 4220 . . . . . . . . . . . . . 14 (𝑥 = 𝐶 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐶, 𝐴}, {𝐶, 𝐵}})
151150sseq1d 3595 . . . . . . . . . . . . 13 (𝑥 = 𝐶 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸))
152 eqeq1 2614 . . . . . . . . . . . . . 14 (𝑥 = 𝐶 → (𝑥 = 𝑦𝐶 = 𝑦))
153152imbi2d 329 . . . . . . . . . . . . 13 (𝑥 = 𝐶 → (((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦) ↔ ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦)))
154151, 153imbi12d 333 . . . . . . . . . . . 12 (𝑥 = 𝐶 → (({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)) ↔ ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦))))
155154imbi2d 329 . . . . . . . . . . 11 (𝑥 = 𝐶 → (({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦))) ↔ ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝐶, 𝐴}, {𝐶, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝐶 = 𝑦)))))
156147, 155syl5ibr 235 . . . . . . . . . 10 (𝑥 = 𝐶 → ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)))))
15763, 110, 1563jaoi 1383 . . . . . . . . 9 ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) → ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)))))
158157com3l 87 . . . . . . . 8 ((𝑦 = 𝐴𝑦 = 𝐵𝑦 = 𝐶) → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)))))
1594, 158sylbi 206 . . . . . . 7 (𝑦 ∈ {𝐴, 𝐵, 𝐶} → ({{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸 → ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦)))))
160159imp 444 . . . . . 6 ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸) → ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦))))
161160com3l 87 . . . . 5 ((𝑥 = 𝐴𝑥 = 𝐵𝑥 = 𝐶) → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸) → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦))))
1622, 161sylbi 206 . . . 4 (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ((𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸) → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦))))
163162imp31 447 . . 3 (((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸)) → ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → 𝑥 = 𝑦))
164163com12 32 . 2 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ {𝐴, 𝐵, 𝐶} USGrph 𝐸) → (((𝑥 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸) ∧ (𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ {{𝑦, 𝐴}, {𝑦, 𝐵}} ⊆ ran 𝐸)) → 𝑥 = 𝑦))
165164alrimivv 1843 1 ((((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))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ⊆ 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:  frgra3vlem2  26528
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