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Theorem cplgr3v 26622
Description: A pseudograph with three (different) vertices is complete iff there is an edge between each of these three vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 5-Nov-2020.) (Proof shortened by AV, 13-Feb-2022.)
Hypotheses
Ref Expression
cplgr3v.e 𝐸 = (Edg‘𝐺)
cplgr3v.t (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶}
Assertion
Ref Expression
cplgr3v (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐺 ∈ ComplGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))

Proof of Theorem cplgr3v
Dummy variables 𝑛 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cplgr3v.t . . . . 5 (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶}
21eqcomi 2774 . . . 4 {𝐴, 𝐵, 𝐶} = (Vtx‘𝐺)
32iscplgrnb 26603 . . 3 (𝐺 ∈ UPGraph → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ {𝐴, 𝐵, 𝐶}∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
433ad2ant2 1164 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐺 ∈ ComplGraph ↔ ∀𝑣 ∈ {𝐴, 𝐵, 𝐶}∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
5 sneq 4344 . . . . . 6 (𝑣 = 𝐴 → {𝑣} = {𝐴})
65difeq2d 3890 . . . . 5 (𝑣 = 𝐴 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐴}))
7 tprot 4439 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
87difeq1i 3886 . . . . . . 7 ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = ({𝐵, 𝐶, 𝐴} ∖ {𝐴})
9 necom 2990 . . . . . . . . 9 (𝐴𝐵𝐵𝐴)
10 necom 2990 . . . . . . . . 9 (𝐴𝐶𝐶𝐴)
11 diftpsn3 4487 . . . . . . . . 9 ((𝐵𝐴𝐶𝐴) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
129, 10, 11syl2anb 591 . . . . . . . 8 ((𝐴𝐵𝐴𝐶) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
13123adant3 1162 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐵, 𝐶, 𝐴} ∖ {𝐴}) = {𝐵, 𝐶})
148, 13syl5eq 2811 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = {𝐵, 𝐶})
15143ad2ant3 1165 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} ∖ {𝐴}) = {𝐵, 𝐶})
166, 15sylan9eqr 2821 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐴) → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = {𝐵, 𝐶})
17 oveq2 6850 . . . . . 6 (𝑣 = 𝐴 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝐴))
1817eleq2d 2830 . . . . 5 (𝑣 = 𝐴 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐴)))
1918adantl 473 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐴) → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐴)))
2016, 19raleqbidv 3300 . . 3 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐴) → (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴)))
21 sneq 4344 . . . . . 6 (𝑣 = 𝐵 → {𝑣} = {𝐵})
2221difeq2d 3890 . . . . 5 (𝑣 = 𝐵 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐵}))
23 tprot 4439 . . . . . . . . 9 {𝐶, 𝐴, 𝐵} = {𝐴, 𝐵, 𝐶}
2423eqcomi 2774 . . . . . . . 8 {𝐴, 𝐵, 𝐶} = {𝐶, 𝐴, 𝐵}
2524difeq1i 3886 . . . . . . 7 ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = ({𝐶, 𝐴, 𝐵} ∖ {𝐵})
26 necom 2990 . . . . . . . . . . . 12 (𝐵𝐶𝐶𝐵)
2726biimpi 207 . . . . . . . . . . 11 (𝐵𝐶𝐶𝐵)
2827anim2i 610 . . . . . . . . . 10 ((𝐴𝐵𝐵𝐶) → (𝐴𝐵𝐶𝐵))
2928ancomd 453 . . . . . . . . 9 ((𝐴𝐵𝐵𝐶) → (𝐶𝐵𝐴𝐵))
30 diftpsn3 4487 . . . . . . . . 9 ((𝐶𝐵𝐴𝐵) → ({𝐶, 𝐴, 𝐵} ∖ {𝐵}) = {𝐶, 𝐴})
3129, 30syl 17 . . . . . . . 8 ((𝐴𝐵𝐵𝐶) → ({𝐶, 𝐴, 𝐵} ∖ {𝐵}) = {𝐶, 𝐴})
32313adant2 1161 . . . . . . 7 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐶, 𝐴, 𝐵} ∖ {𝐵}) = {𝐶, 𝐴})
3325, 32syl5eq 2811 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = {𝐶, 𝐴})
34333ad2ant3 1165 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} ∖ {𝐵}) = {𝐶, 𝐴})
3522, 34sylan9eqr 2821 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐵) → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = {𝐶, 𝐴})
36 oveq2 6850 . . . . . 6 (𝑣 = 𝐵 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝐵))
3736eleq2d 2830 . . . . 5 (𝑣 = 𝐵 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐵)))
3837adantl 473 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐵) → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐵)))
3935, 38raleqbidv 3300 . . 3 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐵) → (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵)))
40 sneq 4344 . . . . . 6 (𝑣 = 𝐶 → {𝑣} = {𝐶})
4140difeq2d 3890 . . . . 5 (𝑣 = 𝐶 → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = ({𝐴, 𝐵, 𝐶} ∖ {𝐶}))
42 diftpsn3 4487 . . . . . . 7 ((𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
43423adant1 1160 . . . . . 6 ((𝐴𝐵𝐴𝐶𝐵𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
44433ad2ant3 1165 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ({𝐴, 𝐵, 𝐶} ∖ {𝐶}) = {𝐴, 𝐵})
4541, 44sylan9eqr 2821 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐶) → ({𝐴, 𝐵, 𝐶} ∖ {𝑣}) = {𝐴, 𝐵})
46 oveq2 6850 . . . . . 6 (𝑣 = 𝐶 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝐶))
4746eleq2d 2830 . . . . 5 (𝑣 = 𝐶 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐶)))
4847adantl 473 . . . 4 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐶) → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝐶)))
4945, 48raleqbidv 3300 . . 3 ((((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) ∧ 𝑣 = 𝐶) → (∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶)))
50 simp1 1166 . . . 4 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐴𝑋)
51503ad2ant1 1163 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → 𝐴𝑋)
52 simp2 1167 . . . 4 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐵𝑌)
53523ad2ant1 1163 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → 𝐵𝑌)
54 simp3 1168 . . . 4 ((𝐴𝑋𝐵𝑌𝐶𝑍) → 𝐶𝑍)
55543ad2ant1 1163 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → 𝐶𝑍)
5620, 39, 49, 51, 53, 55raltpd 4468 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (∀𝑣 ∈ {𝐴, 𝐵, 𝐶}∀𝑛 ∈ ({𝐴, 𝐵, 𝐶} ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ (∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ∧ ∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ∧ ∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶))))
57 eleq1 2832 . . . . . . 7 (𝑛 = 𝐵 → (𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐴)))
58 eleq1 2832 . . . . . . 7 (𝑛 = 𝐶 → (𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
5957, 58ralprg 4390 . . . . . 6 ((𝐵𝑌𝐶𝑍) → (∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴))))
60593adant1 1160 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ↔ (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴))))
61 eleq1 2832 . . . . . . . 8 (𝑛 = 𝐶 → (𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐵)))
62 eleq1 2832 . . . . . . . 8 (𝑛 = 𝐴 → (𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ 𝐴 ∈ (𝐺 NeighbVtx 𝐵)))
6361, 62ralprg 4390 . . . . . . 7 ((𝐶𝑍𝐴𝑋) → (∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵))))
6463ancoms 450 . . . . . 6 ((𝐴𝑋𝐶𝑍) → (∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵))))
65643adant2 1161 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵))))
66 eleq1 2832 . . . . . . 7 (𝑛 = 𝐴 → (𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)))
67 eleq1 2832 . . . . . . 7 (𝑛 = 𝐵 → (𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))
6866, 67ralprg 4390 . . . . . 6 ((𝐴𝑋𝐵𝑌) → (∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))))
69683adant3 1162 . . . . 5 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶) ↔ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))))
7060, 65, 693anbi123d 1560 . . . 4 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ∧ ∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ∧ ∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶)) ↔ ((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵)) ∧ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))))
71703ad2ant1 1163 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ∧ ∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ∧ ∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶)) ↔ ((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵)) ∧ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))))
72 3an6 1570 . . . 4 (((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵)) ∧ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ ((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))))
7372a1i 11 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵)) ∧ (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ ((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))))
74 nbgrsym 26542 . . . . . . 7 (𝐵 ∈ (𝐺 NeighbVtx 𝐴) ↔ 𝐴 ∈ (𝐺 NeighbVtx 𝐵))
75 nbgrsym 26542 . . . . . . 7 (𝐶 ∈ (𝐺 NeighbVtx 𝐵) ↔ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))
76 nbgrsym 26542 . . . . . . 7 (𝐴 ∈ (𝐺 NeighbVtx 𝐶) ↔ 𝐶 ∈ (𝐺 NeighbVtx 𝐴))
7774, 75, 763anbi123i 1194 . . . . . 6 ((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)) ↔ (𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
7877a1i 11 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)) ↔ (𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴))))
7978anbi1d 623 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ ((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))))
80 3anrot 1122 . . . . . . . 8 ((𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)) ↔ (𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)))
8180bicomi 215 . . . . . . 7 ((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))
8281anbi1i 617 . . . . . 6 (((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ ((𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))))
83 anidm 560 . . . . . 6 (((𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))
8482, 83bitri 266 . . . . 5 (((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)))
8584a1i 11 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))))
86 tpid3g 4460 . . . . . . . . 9 (𝐴𝑋𝐴 ∈ {𝐵, 𝐶, 𝐴})
8786, 7syl6eleqr 2855 . . . . . . . 8 (𝐴𝑋𝐴 ∈ {𝐴, 𝐵, 𝐶})
88 tpid3g 4460 . . . . . . . . 9 (𝐵𝑌𝐵 ∈ {𝐶, 𝐴, 𝐵})
8988, 24syl6eleqr 2855 . . . . . . . 8 (𝐵𝑌𝐵 ∈ {𝐴, 𝐵, 𝐶})
90 tpid3g 4460 . . . . . . . 8 (𝐶𝑍𝐶 ∈ {𝐴, 𝐵, 𝐶})
9187, 89, 903anim123i 1190 . . . . . . 7 ((𝐴𝑋𝐵𝑌𝐶𝑍) → (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
92 df-3an 1109 . . . . . . 7 ((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ↔ ((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
9391, 92sylib 209 . . . . . 6 ((𝐴𝑋𝐵𝑌𝐶𝑍) → ((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
94 simplr 785 . . . . . . . . . . 11 (((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) → 𝐵 ∈ {𝐴, 𝐵, 𝐶})
9594anim1i 608 . . . . . . . . . 10 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph) → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ UPGraph))
9695ancomd 453 . . . . . . . . 9 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph) → (𝐺 ∈ UPGraph ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
97963adant3 1162 . . . . . . . 8 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐺 ∈ UPGraph ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}))
98 simpll 783 . . . . . . . . 9 (((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) → 𝐴 ∈ {𝐴, 𝐵, 𝐶})
99 simp1 1166 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝐴𝐵)
10098, 99anim12i 606 . . . . . . . 8 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐴𝐵))
101 cplgr3v.e . . . . . . . . 9 𝐸 = (Edg‘𝐺)
1022, 101nbupgrel 26520 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐴𝐵)) → (𝐴 ∈ (𝐺 NeighbVtx 𝐵) ↔ {𝐴, 𝐵} ∈ 𝐸))
10397, 100, 1023imp3i2an 1452 . . . . . . 7 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐴 ∈ (𝐺 NeighbVtx 𝐵) ↔ {𝐴, 𝐵} ∈ 𝐸))
104 simpr 477 . . . . . . . . . . 11 (((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) → 𝐶 ∈ {𝐴, 𝐵, 𝐶})
105104anim1i 608 . . . . . . . . . 10 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph) → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ UPGraph))
106105ancomd 453 . . . . . . . . 9 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph) → (𝐺 ∈ UPGraph ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
1071063adant3 1162 . . . . . . . 8 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐺 ∈ UPGraph ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}))
108 simp3 1168 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝐵𝐶)
10994, 108anim12i 606 . . . . . . . 8 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵𝐶))
1102, 101nbupgrel 26520 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐵 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵𝐶)) → (𝐵 ∈ (𝐺 NeighbVtx 𝐶) ↔ {𝐵, 𝐶} ∈ 𝐸))
111107, 109, 1103imp3i2an 1452 . . . . . . 7 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐵 ∈ (𝐺 NeighbVtx 𝐶) ↔ {𝐵, 𝐶} ∈ 𝐸))
11298anim1i 608 . . . . . . . . . 10 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph) → (𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ UPGraph))
113112ancomd 453 . . . . . . . . 9 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph) → (𝐺 ∈ UPGraph ∧ 𝐴 ∈ {𝐴, 𝐵, 𝐶}))
1141133adant3 1162 . . . . . . . 8 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐺 ∈ UPGraph ∧ 𝐴 ∈ {𝐴, 𝐵, 𝐶}))
115 simp2 1167 . . . . . . . . . 10 ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝐴𝐶)
116115necomd 2992 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶𝐵𝐶) → 𝐶𝐴)
117104, 116anim12i 606 . . . . . . . 8 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝐴))
1182, 101nbupgrel 26520 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ 𝐴 ∈ {𝐴, 𝐵, 𝐶}) ∧ (𝐶 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐶𝐴)) → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐶, 𝐴} ∈ 𝐸))
119114, 117, 1183imp3i2an 1452 . . . . . . 7 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ↔ {𝐶, 𝐴} ∈ 𝐸))
120103, 111, 1193anbi123d 1560 . . . . . 6 ((((𝐴 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝐵 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐶 ∈ {𝐴, 𝐵, 𝐶}) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
12193, 120syl3an1 1202 . . . . 5 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐴)) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
12280, 121syl5bb 274 . . . 4 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶)) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
12379, 85, 1223bitrd 296 . . 3 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (((𝐵 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐶 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐶)) ∧ (𝐶 ∈ (𝐺 NeighbVtx 𝐴) ∧ 𝐴 ∈ (𝐺 NeighbVtx 𝐵) ∧ 𝐵 ∈ (𝐺 NeighbVtx 𝐶))) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
12471, 73, 1233bitrd 296 . 2 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → ((∀𝑛 ∈ {𝐵, 𝐶}𝑛 ∈ (𝐺 NeighbVtx 𝐴) ∧ ∀𝑛 ∈ {𝐶, 𝐴}𝑛 ∈ (𝐺 NeighbVtx 𝐵) ∧ ∀𝑛 ∈ {𝐴, 𝐵}𝑛 ∈ (𝐺 NeighbVtx 𝐶)) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
1254, 56, 1243bitrd 296 1 (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐺 ∈ ComplGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  cdif 3729  {csn 4334  {cpr 4336  {ctp 4338  cfv 6068  (class class class)co 6842  Vtxcvtx 26165  Edgcedg 26216  UPGraphcupgr 26252   NeighbVtx cnbgr 26503  ComplGraphccplgr 26595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-n0 11539  df-xnn0 11611  df-z 11625  df-uz 11887  df-fz 12534  df-hash 13322  df-edg 26217  df-upgr 26254  df-nbgr 26504  df-uvtx 26567  df-cplgr 26597
This theorem is referenced by:  cusgr3vnbpr  26623
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