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Theorem frgra2v 26526
 Description: Any graph with two (different) vertices is not a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Proof shortened by Alexander van der Vekens, 13-Sep-2018.)
Assertion
Ref Expression
frgra2v (((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) → ¬ {𝐴, 𝐵} FriendGrph 𝐸)

Proof of Theorem frgra2v
Dummy variables 𝑘 𝑙 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neirr 2791 . . . . . . . . . . . . . . . 16 ¬ 𝐴𝐴
2 usgraedgrn 25910 . . . . . . . . . . . . . . . . 17 (({𝐴, 𝐵} USGrph 𝐸 ∧ {𝐴, 𝐴} ∈ ran 𝐸) → 𝐴𝐴)
32ex 449 . . . . . . . . . . . . . . . 16 ({𝐴, 𝐵} USGrph 𝐸 → ({𝐴, 𝐴} ∈ ran 𝐸𝐴𝐴))
41, 3mtoi 189 . . . . . . . . . . . . . . 15 ({𝐴, 𝐵} USGrph 𝐸 → ¬ {𝐴, 𝐴} ∈ ran 𝐸)
54adantl 481 . . . . . . . . . . . . . 14 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ {𝐴, 𝐴} ∈ ran 𝐸)
65intnanrd 954 . . . . . . . . . . . . 13 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ ({𝐴, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸))
7 prex 4836 . . . . . . . . . . . . . 14 {𝐴, 𝐴} ∈ V
8 prex 4836 . . . . . . . . . . . . . 14 {𝐴, 𝐵} ∈ V
97, 8prss 4291 . . . . . . . . . . . . 13 (({𝐴, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸)
106, 9sylnib 317 . . . . . . . . . . . 12 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸)
11 neirr 2791 . . . . . . . . . . . . . . . 16 ¬ 𝐵𝐵
12 usgraedgrn 25910 . . . . . . . . . . . . . . . . 17 (({𝐴, 𝐵} USGrph 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) → 𝐵𝐵)
1312ex 449 . . . . . . . . . . . . . . . 16 ({𝐴, 𝐵} USGrph 𝐸 → ({𝐵, 𝐵} ∈ ran 𝐸𝐵𝐵))
1411, 13mtoi 189 . . . . . . . . . . . . . . 15 ({𝐴, 𝐵} USGrph 𝐸 → ¬ {𝐵, 𝐵} ∈ ran 𝐸)
1514adantl 481 . . . . . . . . . . . . . 14 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ {𝐵, 𝐵} ∈ ran 𝐸)
1615intnand 953 . . . . . . . . . . . . 13 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸))
17 prex 4836 . . . . . . . . . . . . . 14 {𝐵, 𝐴} ∈ V
18 prex 4836 . . . . . . . . . . . . . 14 {𝐵, 𝐵} ∈ V
1917, 18prss 4291 . . . . . . . . . . . . 13 (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐵} ∈ ran 𝐸) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸)
2016, 19sylnib 317 . . . . . . . . . . . 12 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸)
21 ioran 510 . . . . . . . . . . . 12 (¬ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸) ↔ (¬ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 ∧ ¬ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸))
2210, 20, 21sylanbrc 695 . . . . . . . . . . 11 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸))
23 preq1 4212 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → {𝑥, 𝐴} = {𝐴, 𝐴})
24 preq1 4212 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
2523, 24preq12d 4220 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
2625sseq1d 3595 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸))
27 preq1 4212 . . . . . . . . . . . . . . 15 (𝑥 = 𝐵 → {𝑥, 𝐴} = {𝐵, 𝐴})
28 preq1 4212 . . . . . . . . . . . . . . 15 (𝑥 = 𝐵 → {𝑥, 𝐵} = {𝐵, 𝐵})
2927, 28preq12d 4220 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
3029sseq1d 3595 . . . . . . . . . . . . 13 (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸))
3126, 30rexprg 4182 . . . . . . . . . . . 12 ((𝐴𝑋𝐵𝑌) → (∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸)))
3231ad2antrr 758 . . . . . . . . . . 11 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ ran 𝐸 ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ ran 𝐸)))
3322, 32mtbird 314 . . . . . . . . . 10 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ ∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸)
34 reurex 3137 . . . . . . . . . 10 (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 → ∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸)
3533, 34nsyl 134 . . . . . . . . 9 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸)
3635orcd 406 . . . . . . . 8 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ∨ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
37 rexnal 2978 . . . . . . . . . . . 12 (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸)
3837bicomi 213 . . . . . . . . . . 11 (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸)
3938a1i 11 . . . . . . . . . 10 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
40 difprsn1 4271 . . . . . . . . . . . 12 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
4140ad2antlr 759 . . . . . . . . . . 11 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
4241rexeqdv 3122 . . . . . . . . . 10 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃𝑙 ∈ {𝐵} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
43 preq2 4213 . . . . . . . . . . . . . . . 16 (𝑙 = 𝐵 → {𝑥, 𝑙} = {𝑥, 𝐵})
4443preq2d 4219 . . . . . . . . . . . . . . 15 (𝑙 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝑙}} = {{𝑥, 𝐴}, {𝑥, 𝐵}})
4544sseq1d 3595 . . . . . . . . . . . . . 14 (𝑙 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
4645reubidv 3103 . . . . . . . . . . . . 13 (𝑙 = 𝐵 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
4746notbid 307 . . . . . . . . . . . 12 (𝑙 = 𝐵 → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
4847rexsng 4166 . . . . . . . . . . 11 (𝐵𝑌 → (∃𝑙 ∈ {𝐵} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
4948ad3antlr 763 . . . . . . . . . 10 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (∃𝑙 ∈ {𝐵} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
5039, 42, 493bitrd 293 . . . . . . . . 9 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸))
51 rexnal 2978 . . . . . . . . . . . 12 (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸)
5251bicomi 213 . . . . . . . . . . 11 (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸)
5352a1i 11 . . . . . . . . . 10 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
54 difprsn2 4272 . . . . . . . . . . . 12 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
5554ad2antlr 759 . . . . . . . . . . 11 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
5655rexeqdv 3122 . . . . . . . . . 10 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃𝑙 ∈ {𝐴} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
57 preq2 4213 . . . . . . . . . . . . . . . 16 (𝑙 = 𝐴 → {𝑥, 𝑙} = {𝑥, 𝐴})
5857preq2d 4219 . . . . . . . . . . . . . . 15 (𝑙 = 𝐴 → {{𝑥, 𝐵}, {𝑥, 𝑙}} = {{𝑥, 𝐵}, {𝑥, 𝐴}})
5958sseq1d 3595 . . . . . . . . . . . . . 14 (𝑙 = 𝐴 → ({{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
6059reubidv 3103 . . . . . . . . . . . . 13 (𝑙 = 𝐴 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
6160notbid 307 . . . . . . . . . . . 12 (𝑙 = 𝐴 → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
6261rexsng 4166 . . . . . . . . . . 11 (𝐴𝑋 → (∃𝑙 ∈ {𝐴} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
6362ad3antrrr 762 . . . . . . . . . 10 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (∃𝑙 ∈ {𝐴} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
6453, 56, 633bitrd 293 . . . . . . . . 9 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸))
6550, 64orbi12d 742 . . . . . . . 8 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ((¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸) ↔ (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ ran 𝐸 ∨ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ ran 𝐸)))
6636, 65mpbird 246 . . . . . . 7 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
67 sneq 4135 . . . . . . . . . . . 12 (𝑘 = 𝐴 → {𝑘} = {𝐴})
6867difeq2d 3690 . . . . . . . . . . 11 (𝑘 = 𝐴 → ({𝐴, 𝐵} ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝐴}))
69 preq2 4213 . . . . . . . . . . . . . 14 (𝑘 = 𝐴 → {𝑥, 𝑘} = {𝑥, 𝐴})
7069preq1d 4218 . . . . . . . . . . . . 13 (𝑘 = 𝐴 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝐴}, {𝑥, 𝑙}})
7170sseq1d 3595 . . . . . . . . . . . 12 (𝑘 = 𝐴 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
7271reubidv 3103 . . . . . . . . . . 11 (𝑘 = 𝐴 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
7368, 72raleqbidv 3129 . . . . . . . . . 10 (𝑘 = 𝐴 → (∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
7473notbid 307 . . . . . . . . 9 (𝑘 = 𝐴 → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸))
75 sneq 4135 . . . . . . . . . . . 12 (𝑘 = 𝐵 → {𝑘} = {𝐵})
7675difeq2d 3690 . . . . . . . . . . 11 (𝑘 = 𝐵 → ({𝐴, 𝐵} ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝐵}))
77 preq2 4213 . . . . . . . . . . . . . 14 (𝑘 = 𝐵 → {𝑥, 𝑘} = {𝑥, 𝐵})
7877preq1d 4218 . . . . . . . . . . . . 13 (𝑘 = 𝐵 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝐵}, {𝑥, 𝑙}})
7978sseq1d 3595 . . . . . . . . . . . 12 (𝑘 = 𝐵 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
8079reubidv 3103 . . . . . . . . . . 11 (𝑘 = 𝐵 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
8176, 80raleqbidv 3129 . . . . . . . . . 10 (𝑘 = 𝐵 → (∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
8281notbid 307 . . . . . . . . 9 (𝑘 = 𝐵 → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸))
8374, 82rexprg 4182 . . . . . . . 8 ((𝐴𝑋𝐵𝑌) → (∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
8483ad2antrr 758 . . . . . . 7 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → (∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ ran 𝐸 ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
8566, 84mpbird 246 . . . . . 6 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)
86 rexnal 2978 . . . . . 6 (∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ¬ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)
8785, 86sylib 207 . . . . 5 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)
8887intnand 953 . . . 4 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ ({𝐴, 𝐵} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))
89 usgrav 25867 . . . . . 6 ({𝐴, 𝐵} USGrph 𝐸 → ({𝐴, 𝐵} ∈ V ∧ 𝐸 ∈ V))
9089adantl 481 . . . . 5 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ({𝐴, 𝐵} ∈ V ∧ 𝐸 ∈ V))
91 isfrgra 26517 . . . . 5 (({𝐴, 𝐵} ∈ V ∧ 𝐸 ∈ V) → ({𝐴, 𝐵} FriendGrph 𝐸 ↔ ({𝐴, 𝐵} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
9290, 91syl 17 . . . 4 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ({𝐴, 𝐵} FriendGrph 𝐸 ↔ ({𝐴, 𝐵} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
9388, 92mtbird 314 . . 3 ((((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) ∧ {𝐴, 𝐵} USGrph 𝐸) → ¬ {𝐴, 𝐵} FriendGrph 𝐸)
9493expcom 450 . 2 ({𝐴, 𝐵} USGrph 𝐸 → (((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) → ¬ {𝐴, 𝐵} FriendGrph 𝐸))
95 frisusgra 26519 . . . 4 ({𝐴, 𝐵} FriendGrph 𝐸 → {𝐴, 𝐵} USGrph 𝐸)
9695con3i 149 . . 3 (¬ {𝐴, 𝐵} USGrph 𝐸 → ¬ {𝐴, 𝐵} FriendGrph 𝐸)
9796a1d 25 . 2 (¬ {𝐴, 𝐵} USGrph 𝐸 → (((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) → ¬ {𝐴, 𝐵} FriendGrph 𝐸))
9894, 97pm2.61i 175 1 (((𝐴𝑋𝐵𝑌) ∧ 𝐴𝐵) → ¬ {𝐴, 𝐵} FriendGrph 𝐸)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039   USGrph cusg 25859   FriendGrph cfrgra 26515 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  df-frgra 26516 This theorem is referenced by:  1to2vfriswmgra  26533  frgraregord013  26645
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