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Theorem nfrgr2v 27034
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.) (Revised by AV, 29-Mar-2021.)
Assertion
Ref Expression
nfrgr2v (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph )

Proof of Theorem nfrgr2v
Dummy variables 𝑘 𝑙 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neirr 2799 . . . . . . . . . . . . . . . . . 18 ¬ 𝐴𝐴
2 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 (Edg‘𝐺) = (Edg‘𝐺)
32usgredgne 26025 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ USGraph ∧ {𝐴, 𝐴} ∈ (Edg‘𝐺)) → 𝐴𝐴)
43ex 450 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ USGraph → ({𝐴, 𝐴} ∈ (Edg‘𝐺) → 𝐴𝐴))
51, 4mtoi 190 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ USGraph → ¬ {𝐴, 𝐴} ∈ (Edg‘𝐺))
65adantl 482 . . . . . . . . . . . . . . . 16 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ {𝐴, 𝐴} ∈ (Edg‘𝐺))
76intnanrd 962 . . . . . . . . . . . . . . 15 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ ({𝐴, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐵} ∈ (Edg‘𝐺)))
8 prex 4880 . . . . . . . . . . . . . . . 16 {𝐴, 𝐴} ∈ V
9 prex 4880 . . . . . . . . . . . . . . . 16 {𝐴, 𝐵} ∈ V
108, 9prss 4326 . . . . . . . . . . . . . . 15 (({𝐴, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐴, 𝐵} ∈ (Edg‘𝐺)) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺))
117, 10sylnib 318 . . . . . . . . . . . . . 14 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺))
12 neirr 2799 . . . . . . . . . . . . . . . . . 18 ¬ 𝐵𝐵
132usgredgne 26025 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ USGraph ∧ {𝐵, 𝐵} ∈ (Edg‘𝐺)) → 𝐵𝐵)
1413ex 450 . . . . . . . . . . . . . . . . . 18 (𝐺 ∈ USGraph → ({𝐵, 𝐵} ∈ (Edg‘𝐺) → 𝐵𝐵))
1512, 14mtoi 190 . . . . . . . . . . . . . . . . 17 (𝐺 ∈ USGraph → ¬ {𝐵, 𝐵} ∈ (Edg‘𝐺))
1615adantl 482 . . . . . . . . . . . . . . . 16 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ {𝐵, 𝐵} ∈ (Edg‘𝐺))
1716intnand 961 . . . . . . . . . . . . . . 15 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ ({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐵} ∈ (Edg‘𝐺)))
18 prex 4880 . . . . . . . . . . . . . . . 16 {𝐵, 𝐴} ∈ V
19 prex 4880 . . . . . . . . . . . . . . . 16 {𝐵, 𝐵} ∈ V
2018, 19prss 4326 . . . . . . . . . . . . . . 15 (({𝐵, 𝐴} ∈ (Edg‘𝐺) ∧ {𝐵, 𝐵} ∈ (Edg‘𝐺)) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺))
2117, 20sylnib 318 . . . . . . . . . . . . . 14 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺))
22 ioran 511 . . . . . . . . . . . . . 14 (¬ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺) ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺)) ↔ (¬ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺) ∧ ¬ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺)))
2311, 21, 22sylanbrc 697 . . . . . . . . . . . . 13 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺) ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺)))
24 preq1 4245 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐴 → {𝑥, 𝐴} = {𝐴, 𝐴})
25 preq1 4245 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
2624, 25preq12d 4253 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐴 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐴, 𝐴}, {𝐴, 𝐵}})
2726sseq1d 3617 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ↔ {{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺)))
28 preq1 4245 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐵 → {𝑥, 𝐴} = {𝐵, 𝐴})
29 preq1 4245 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝐵 → {𝑥, 𝐵} = {𝐵, 𝐵})
3028, 29preq12d 4253 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝐵}} = {{𝐵, 𝐴}, {𝐵, 𝐵}})
3130sseq1d 3617 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ↔ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺)))
3227, 31rexprg 4213 . . . . . . . . . . . . . . 15 ((𝐴𝑋𝐵𝑌) → (∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺) ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺))))
33323adant3 1079 . . . . . . . . . . . . . 14 ((𝐴𝑋𝐵𝑌𝐴𝐵) → (∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺) ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺))))
3433adantr 481 . . . . . . . . . . . . 13 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → (∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ↔ ({{𝐴, 𝐴}, {𝐴, 𝐵}} ⊆ (Edg‘𝐺) ∨ {{𝐵, 𝐴}, {𝐵, 𝐵}} ⊆ (Edg‘𝐺))))
3523, 34mtbird 315 . . . . . . . . . . . 12 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ ∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺))
36 reurex 3153 . . . . . . . . . . . 12 (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) → ∃𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺))
3735, 36nsyl 135 . . . . . . . . . . 11 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺))
3837orcd 407 . . . . . . . . . 10 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ∨ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
39 rexnal 2991 . . . . . . . . . . . . . 14 (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
4039bicomi 214 . . . . . . . . . . . . 13 (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
4140a1i 11 . . . . . . . . . . . 12 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
42 difprsn1 4306 . . . . . . . . . . . . . . 15 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
43423ad2ant3 1082 . . . . . . . . . . . . . 14 ((𝐴𝑋𝐵𝑌𝐴𝐵) → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
4443adantr 481 . . . . . . . . . . . . 13 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
4544rexeqdv 3138 . . . . . . . . . . . 12 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃𝑙 ∈ {𝐵} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
46 preq2 4246 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝐵 → {𝑥, 𝑙} = {𝑥, 𝐵})
4746preq2d 4252 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝐵 → {{𝑥, 𝐴}, {𝑥, 𝑙}} = {{𝑥, 𝐴}, {𝑥, 𝐵}})
4847sseq1d 3617 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝐵 → ({{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
4948reubidv 3119 . . . . . . . . . . . . . . . 16 (𝑙 = 𝐵 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
5049notbid 308 . . . . . . . . . . . . . . 15 (𝑙 = 𝐵 → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
5150rexsng 4197 . . . . . . . . . . . . . 14 (𝐵𝑌 → (∃𝑙 ∈ {𝐵} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
52513ad2ant2 1081 . . . . . . . . . . . . 13 ((𝐴𝑋𝐵𝑌𝐴𝐵) → (∃𝑙 ∈ {𝐵} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
5352adantr 481 . . . . . . . . . . . 12 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → (∃𝑙 ∈ {𝐵} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
5441, 45, 533bitrd 294 . . . . . . . . . . 11 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺)))
55 rexnal 2991 . . . . . . . . . . . . . 14 (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
5655bicomi 214 . . . . . . . . . . . . 13 (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
5756a1i 11 . . . . . . . . . . . 12 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
58 difprsn2 4307 . . . . . . . . . . . . . . 15 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
59583ad2ant3 1082 . . . . . . . . . . . . . 14 ((𝐴𝑋𝐵𝑌𝐴𝐵) → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
6059adantr 481 . . . . . . . . . . . . 13 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
6160rexeqdv 3138 . . . . . . . . . . . 12 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → (∃𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵}) ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃𝑙 ∈ {𝐴} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
62 preq2 4246 . . . . . . . . . . . . . . . . . . 19 (𝑙 = 𝐴 → {𝑥, 𝑙} = {𝑥, 𝐴})
6362preq2d 4252 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝐴 → {{𝑥, 𝐵}, {𝑥, 𝑙}} = {{𝑥, 𝐵}, {𝑥, 𝐴}})
6463sseq1d 3617 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝐴 → ({{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
6564reubidv 3119 . . . . . . . . . . . . . . . 16 (𝑙 = 𝐴 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
6665notbid 308 . . . . . . . . . . . . . . 15 (𝑙 = 𝐴 → (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
6766rexsng 4197 . . . . . . . . . . . . . 14 (𝐴𝑋 → (∃𝑙 ∈ {𝐴} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
68673ad2ant1 1080 . . . . . . . . . . . . 13 ((𝐴𝑋𝐵𝑌𝐴𝐵) → (∃𝑙 ∈ {𝐴} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
6968adantr 481 . . . . . . . . . . . 12 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → (∃𝑙 ∈ {𝐴} ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
7057, 61, 693bitrd 294 . . . . . . . . . . 11 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺)))
7154, 70orbi12d 745 . . . . . . . . . 10 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → ((¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)) ↔ (¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝐵}} ⊆ (Edg‘𝐺) ∨ ¬ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝐴}} ⊆ (Edg‘𝐺))))
7238, 71mpbird 247 . . . . . . . . 9 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
73 sneq 4165 . . . . . . . . . . . . . . 15 (𝑘 = 𝐴 → {𝑘} = {𝐴})
7473difeq2d 3712 . . . . . . . . . . . . . 14 (𝑘 = 𝐴 → ({𝐴, 𝐵} ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝐴}))
75 preq2 4246 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝐴 → {𝑥, 𝑘} = {𝑥, 𝐴})
7675preq1d 4251 . . . . . . . . . . . . . . . 16 (𝑘 = 𝐴 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝐴}, {𝑥, 𝑙}})
7776sseq1d 3617 . . . . . . . . . . . . . . 15 (𝑘 = 𝐴 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
7877reubidv 3119 . . . . . . . . . . . . . 14 (𝑘 = 𝐴 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
7974, 78raleqbidv 3145 . . . . . . . . . . . . 13 (𝑘 = 𝐴 → (∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
8079notbid 308 . . . . . . . . . . . 12 (𝑘 = 𝐴 → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
81 sneq 4165 . . . . . . . . . . . . . . 15 (𝑘 = 𝐵 → {𝑘} = {𝐵})
8281difeq2d 3712 . . . . . . . . . . . . . 14 (𝑘 = 𝐵 → ({𝐴, 𝐵} ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝐵}))
83 preq2 4246 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝐵 → {𝑥, 𝑘} = {𝑥, 𝐵})
8483preq1d 4251 . . . . . . . . . . . . . . . 16 (𝑘 = 𝐵 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝐵}, {𝑥, 𝑙}})
8584sseq1d 3617 . . . . . . . . . . . . . . 15 (𝑘 = 𝐵 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
8685reubidv 3119 . . . . . . . . . . . . . 14 (𝑘 = 𝐵 → (∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
8782, 86raleqbidv 3145 . . . . . . . . . . . . 13 (𝑘 = 𝐵 → (∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
8887notbid 308 . . . . . . . . . . . 12 (𝑘 = 𝐵 → (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
8980, 88rexprg 4213 . . . . . . . . . . 11 ((𝐴𝑋𝐵𝑌) → (∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))))
90893adant3 1079 . . . . . . . . . 10 ((𝐴𝑋𝐵𝑌𝐴𝐵) → (∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))))
9190adantr 481 . . . . . . . . 9 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → (∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ (¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐴})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐴}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ∨ ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝐵})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝐵}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))))
9272, 91mpbird 247 . . . . . . . 8 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → ∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
93 rexnal 2991 . . . . . . . 8 (∃𝑘 ∈ {𝐴, 𝐵} ¬ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ¬ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
9492, 93sylib 208 . . . . . . 7 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))
9594intnand 961 . . . . . 6 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ 𝐺 ∈ USGraph ) → ¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
9695adantlr 750 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) ∧ 𝐺 ∈ USGraph ) → ¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
97 id 22 . . . . . . . . . 10 ((Vtx‘𝐺) = {𝐴, 𝐵} → (Vtx‘𝐺) = {𝐴, 𝐵})
98 difeq1 3705 . . . . . . . . . . 11 ((Vtx‘𝐺) = {𝐴, 𝐵} → ((Vtx‘𝐺) ∖ {𝑘}) = ({𝐴, 𝐵} ∖ {𝑘}))
99 reueq1 3133 . . . . . . . . . . 11 ((Vtx‘𝐺) = {𝐴, 𝐵} → (∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
10098, 99raleqbidv 3145 . . . . . . . . . 10 ((Vtx‘𝐺) = {𝐴, 𝐵} → (∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
10197, 100raleqbidv 3145 . . . . . . . . 9 ((Vtx‘𝐺) = {𝐴, 𝐵} → (∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺) ↔ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
102101anbi2d 739 . . . . . . . 8 ((Vtx‘𝐺) = {𝐴, 𝐵} → ((𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)) ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))))
103102notbid 308 . . . . . . 7 ((Vtx‘𝐺) = {𝐴, 𝐵} → (¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)) ↔ ¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))))
104103adantl 482 . . . . . 6 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → (¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)) ↔ ¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))))
105104adantr 481 . . . . 5 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) ∧ 𝐺 ∈ USGraph ) → (¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)) ↔ ¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ {𝐴, 𝐵}∀𝑙 ∈ ({𝐴, 𝐵} ∖ {𝑘})∃!𝑥 ∈ {𝐴, 𝐵} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺))))
10696, 105mpbird 247 . . . 4 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) ∧ 𝐺 ∈ USGraph ) → ¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
107 df-nel 2894 . . . . 5 (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
108 eqid 2621 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
109108, 2frgrusgrfrcond 27023 . . . . 5 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
110107, 109xchbinx 324 . . . 4 (𝐺 ∉ FriendGraph ↔ ¬ (𝐺 ∈ USGraph ∧ ∀𝑘 ∈ (Vtx‘𝐺)∀𝑙 ∈ ((Vtx‘𝐺) ∖ {𝑘})∃!𝑥 ∈ (Vtx‘𝐺){{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ (Edg‘𝐺)))
111106, 110sylibr 224 . . 3 ((((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) ∧ 𝐺 ∈ USGraph ) → 𝐺 ∉ FriendGraph )
112111expcom 451 . 2 (𝐺 ∈ USGraph → (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph ))
113 frgrusgr 27024 . . . . 5 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
114113con3i 150 . . . 4 𝐺 ∈ USGraph → ¬ 𝐺 ∈ FriendGraph )
115114, 107sylibr 224 . . 3 𝐺 ∈ USGraph → 𝐺 ∉ FriendGraph )
116115a1d 25 . 2 𝐺 ∈ USGraph → (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph ))
117112, 116pm2.61i 176 1 (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (Vtx‘𝐺) = {𝐴, 𝐵}) → 𝐺 ∉ FriendGraph )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wnel 2893  wral 2908  wrex 2909  ∃!wreu 2910  cdif 3557  wss 3560  {csn 4155  {cpr 4157  cfv 5857  Vtxcvtx 25808  Edgcedg 25873   USGraph cusgr 25971   FriendGraph cfrgr 27020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-hash 13074  df-edg 25874  df-umgr 25908  df-usgr 25973  df-frgr 27021
This theorem is referenced by:  1to2vfriswmgr  27041  frgrregord013  27141
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