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Theorem nb3grapr2 25983
 Description: The neighbors of a vertex in a graph with three elements are an unordered pair of the other vertices if and only if all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
Assertion
Ref Expression
nb3grapr2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐴, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵})))

Proof of Theorem nb3grapr2
StepHypRef Expression
1 3anan32 1043 . . . . 5 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ {𝐵, 𝐶} ∈ ran 𝐸))
21a1i 11 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
3 prcom 4211 . . . . . . . . . . 11 {𝐶, 𝐴} = {𝐴, 𝐶}
43eleq1i 2679 . . . . . . . . . 10 ({𝐶, 𝐴} ∈ ran 𝐸 ↔ {𝐴, 𝐶} ∈ ran 𝐸)
54biimpi 205 . . . . . . . . 9 ({𝐶, 𝐴} ∈ ran 𝐸 → {𝐴, 𝐶} ∈ ran 𝐸)
65pm4.71i 662 . . . . . . . 8 ({𝐶, 𝐴} ∈ ran 𝐸 ↔ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸))
76anbi2i 726 . . . . . . 7 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)))
8 anass 679 . . . . . . 7 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ {𝐴, 𝐶} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)))
97, 8bitr4i 266 . . . . . 6 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ {𝐴, 𝐶} ∈ ran 𝐸))
109anbi1i 727 . . . . 5 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ {𝐵, 𝐶} ∈ ran 𝐸) ↔ ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ {𝐵, 𝐶} ∈ ran 𝐸))
11 anass 679 . . . . 5 (((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ {𝐵, 𝐶} ∈ ran 𝐸) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
1210, 11bitri 263 . . . 4 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ {𝐵, 𝐶} ∈ ran 𝐸) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
132, 12syl6bb 275 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
14 prcom 4211 . . . . . . . . . 10 {𝐴, 𝐵} = {𝐵, 𝐴}
1514eleq1i 2679 . . . . . . . . 9 ({𝐴, 𝐵} ∈ ran 𝐸 ↔ {𝐵, 𝐴} ∈ ran 𝐸)
1615biimpi 205 . . . . . . . 8 ({𝐴, 𝐵} ∈ ran 𝐸 → {𝐵, 𝐴} ∈ ran 𝐸)
1716pm4.71i 662 . . . . . . 7 ({𝐴, 𝐵} ∈ ran 𝐸 ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸))
1817anbi1i 727 . . . . . 6 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ {𝐶, 𝐴} ∈ ran 𝐸))
19 df-3an 1033 . . . . . 6 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸) ∧ {𝐶, 𝐴} ∈ ran 𝐸))
2018, 19bitr4i 266 . . . . 5 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸))
21 prcom 4211 . . . . . . . . . 10 {𝐵, 𝐶} = {𝐶, 𝐵}
2221eleq1i 2679 . . . . . . . . 9 ({𝐵, 𝐶} ∈ ran 𝐸 ↔ {𝐶, 𝐵} ∈ ran 𝐸)
2322biimpi 205 . . . . . . . 8 ({𝐵, 𝐶} ∈ ran 𝐸 → {𝐶, 𝐵} ∈ ran 𝐸)
2423pm4.71i 662 . . . . . . 7 ({𝐵, 𝐶} ∈ ran 𝐸 ↔ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))
2524anbi2i 726 . . . . . 6 (({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
26 3anass 1035 . . . . . 6 (({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ ({𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
2725, 26bitr4i 266 . . . . 5 (({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ↔ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))
2820, 27anbi12i 729 . . . 4 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
29 an6 1400 . . . 4 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
3028, 29bitri 263 . . 3 ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ∧ ({𝐴, 𝐶} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
3113, 30syl6bb 275 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸))))
32 nb3graprlem1 25980 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸)))
3332bicomd 212 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ↔ (⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶}))
34 3ancoma 1038 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) ↔ (𝐵𝑌𝐴𝑋𝐶𝑍))
3534biimpi 205 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐵𝑌𝐴𝑋𝐶𝑍))
36 tpcoma 4229 . . . . . . . . 9 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐴, 𝐶}
3736eqeq2i 2622 . . . . . . . 8 (𝑉 = {𝐴, 𝐵, 𝐶} ↔ 𝑉 = {𝐵, 𝐴, 𝐶})
3837biimpi 205 . . . . . . 7 (𝑉 = {𝐴, 𝐵, 𝐶} → 𝑉 = {𝐵, 𝐴, 𝐶})
3938anim1i 590 . . . . . 6 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → (𝑉 = {𝐵, 𝐴, 𝐶} ∧ 𝑉 USGrph 𝐸))
40 nb3graprlem1 25980 . . . . . 6 (((𝐵𝑌𝐴𝑋𝐶𝑍) ∧ (𝑉 = {𝐵, 𝐴, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐴, 𝐶} ↔ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
4135, 39, 40syl2an 493 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐴, 𝐶} ↔ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
4241bicomd 212 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ↔ (⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐴, 𝐶}))
43 3anrot 1036 . . . . . . 7 ((𝐶𝑍𝐴𝑋𝐵𝑌) ↔ (𝐴𝑋𝐵𝑌𝐶𝑍))
4443biimpri 217 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐶𝑍𝐴𝑋𝐵𝑌))
45 tprot 4228 . . . . . . . . . 10 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
4645eqcomi 2619 . . . . . . . . 9 {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
4746eqeq2i 2622 . . . . . . . 8 (𝑉 = {𝐴, 𝐵, 𝐶} ↔ 𝑉 = {𝐶, 𝐴, 𝐵})
4847biimpi 205 . . . . . . 7 (𝑉 = {𝐴, 𝐵, 𝐶} → 𝑉 = {𝐶, 𝐴, 𝐵})
4948anim1i 590 . . . . . 6 ((𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) → (𝑉 = {𝐶, 𝐴, 𝐵} ∧ 𝑉 USGrph 𝐸))
50 nb3graprlem1 25980 . . . . . 6 (((𝐶𝑍𝐴𝑋𝐵𝑌) ∧ (𝑉 = {𝐶, 𝐴, 𝐵} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵} ↔ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
5144, 49, 50syl2an 493 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ((⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵} ↔ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)))
5251bicomd 212 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → (({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸) ↔ (⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵}))
5333, 42, 523anbi123d 1391 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸)) → ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)) ↔ ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐴, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵})))
54533adant3 1074 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐴, 𝐶} ∈ ran 𝐸) ∧ ({𝐵, 𝐴} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) ∧ ({𝐶, 𝐴} ∈ ran 𝐸 ∧ {𝐶, 𝐵} ∈ ran 𝐸)) ↔ ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐴, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵})))
5531, 54bitrd 267 1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ (𝑉 = {𝐴, 𝐵, 𝐶} ∧ 𝑉 USGrph 𝐸) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) ↔ ((⟨𝑉, 𝐸⟩ Neighbors 𝐴) = {𝐵, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐵) = {𝐴, 𝐶} ∧ (⟨𝑉, 𝐸⟩ Neighbors 𝐶) = {𝐴, 𝐵})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {cpr 4127  {ctp 4129  ⟨cop 4131   class class class wbr 4583  ran crn 5039  (class class class)co 6549   USGrph cusg 25859   Neighbors cnbgra 25946 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-nbgra 25949 This theorem is referenced by: (None)
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