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Theorem upgrex 27595
Description: An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
Hypotheses
Ref Expression
isupgr.v 𝑉 = (Vtx‘𝐺)
isupgr.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
upgrex ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → ∃𝑥𝑉𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
Distinct variable groups:   𝑥,𝐺   𝑥,𝑉   𝑥,𝐸   𝑥,𝐹   𝑥,𝐴,𝑦   𝑦,𝐸   𝑦,𝐹   𝑦,𝐺   𝑦,𝑉

Proof of Theorem upgrex
StepHypRef Expression
1 isupgr.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 isupgr.e . . . . 5 𝐸 = (iEdg‘𝐺)
31, 2upgrn0 27592 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ≠ ∅)
4 n0 4290 . . . 4 ((𝐸𝐹) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐸𝐹))
53, 4sylib 217 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → ∃𝑥 𝑥 ∈ (𝐸𝐹))
6 simp1 1135 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → 𝐺 ∈ UPGraph)
7 fndm 6574 . . . . . . . . . . . . 13 (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
87eqcomd 2742 . . . . . . . . . . . 12 (𝐸 Fn 𝐴𝐴 = dom 𝐸)
98eleq2d 2822 . . . . . . . . . . 11 (𝐸 Fn 𝐴 → (𝐹𝐴𝐹 ∈ dom 𝐸))
109biimpd 228 . . . . . . . . . 10 (𝐸 Fn 𝐴 → (𝐹𝐴𝐹 ∈ dom 𝐸))
1110a1i 11 . . . . . . . . 9 (𝐺 ∈ UPGraph → (𝐸 Fn 𝐴 → (𝐹𝐴𝐹 ∈ dom 𝐸)))
12113imp 1110 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → 𝐹 ∈ dom 𝐸)
131, 2upgrss 27591 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)
146, 12, 13syl2anc 584 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ⊆ 𝑉)
1514sselda 3930 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → 𝑥𝑉)
1615adantr 481 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → 𝑥𝑉)
17 simpr 485 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → ((𝐸𝐹) ∖ {𝑥}) = ∅)
18 ssdif0 4307 . . . . . . . . . 10 ((𝐸𝐹) ⊆ {𝑥} ↔ ((𝐸𝐹) ∖ {𝑥}) = ∅)
1917, 18sylibr 233 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → (𝐸𝐹) ⊆ {𝑥})
20 simpr 485 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → 𝑥 ∈ (𝐸𝐹))
2120snssd 4753 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → {𝑥} ⊆ (𝐸𝐹))
2221adantr 481 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → {𝑥} ⊆ (𝐸𝐹))
2319, 22eqssd 3947 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → (𝐸𝐹) = {𝑥})
24 preq2 4679 . . . . . . . . . 10 (𝑦 = 𝑥 → {𝑥, 𝑦} = {𝑥, 𝑥})
25 dfsn2 4583 . . . . . . . . . 10 {𝑥} = {𝑥, 𝑥}
2624, 25eqtr4di 2794 . . . . . . . . 9 (𝑦 = 𝑥 → {𝑥, 𝑦} = {𝑥})
2726rspceeqv 3583 . . . . . . . 8 ((𝑥𝑉 ∧ (𝐸𝐹) = {𝑥}) → ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
2816, 23, 27syl2anc 584 . . . . . . 7 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
29 n0 4290 . . . . . . . 8 (((𝐸𝐹) ∖ {𝑥}) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))
3014adantr 481 . . . . . . . . . . . . . 14 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝐸𝐹) ⊆ 𝑉)
31 simprr 770 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))
3231eldifad 3908 . . . . . . . . . . . . . 14 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦 ∈ (𝐸𝐹))
3330, 32sseldd 3931 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦𝑉)
341, 2upgrfi 27594 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ∈ Fin)
3534adantr 481 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝐸𝐹) ∈ Fin)
36 simprl 768 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑥 ∈ (𝐸𝐹))
3736, 32prssd 4766 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} ⊆ (𝐸𝐹))
38 fvex 6824 . . . . . . . . . . . . . . . . 17 (𝐸𝐹) ∈ V
39 ssdomg 8839 . . . . . . . . . . . . . . . . 17 ((𝐸𝐹) ∈ V → ({𝑥, 𝑦} ⊆ (𝐸𝐹) → {𝑥, 𝑦} ≼ (𝐸𝐹)))
4038, 37, 39mpsyl 68 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} ≼ (𝐸𝐹))
411, 2upgrle 27593 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (♯‘(𝐸𝐹)) ≤ 2)
4241adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (♯‘(𝐸𝐹)) ≤ 2)
43 eldifsni 4734 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}) → 𝑦𝑥)
4443ad2antll 726 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦𝑥)
4544necomd 2996 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑥𝑦)
46 hashprg 14188 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑦 ↔ (♯‘{𝑥, 𝑦}) = 2))
4746el2v 3448 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑦 ↔ (♯‘{𝑥, 𝑦}) = 2)
4845, 47sylib 217 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (♯‘{𝑥, 𝑦}) = 2)
4942, 48breqtrrd 5114 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (♯‘(𝐸𝐹)) ≤ (♯‘{𝑥, 𝑦}))
50 prfi 9165 . . . . . . . . . . . . . . . . . 18 {𝑥, 𝑦} ∈ Fin
51 hashdom 14172 . . . . . . . . . . . . . . . . . 18 (((𝐸𝐹) ∈ Fin ∧ {𝑥, 𝑦} ∈ Fin) → ((♯‘(𝐸𝐹)) ≤ (♯‘{𝑥, 𝑦}) ↔ (𝐸𝐹) ≼ {𝑥, 𝑦}))
5235, 50, 51sylancl 586 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → ((♯‘(𝐸𝐹)) ≤ (♯‘{𝑥, 𝑦}) ↔ (𝐸𝐹) ≼ {𝑥, 𝑦}))
5349, 52mpbid 231 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝐸𝐹) ≼ {𝑥, 𝑦})
54 sbth 8936 . . . . . . . . . . . . . . . 16 (({𝑥, 𝑦} ≼ (𝐸𝐹) ∧ (𝐸𝐹) ≼ {𝑥, 𝑦}) → {𝑥, 𝑦} ≈ (𝐸𝐹))
5540, 53, 54syl2anc 584 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} ≈ (𝐸𝐹))
56 fisseneq 9100 . . . . . . . . . . . . . . 15 (((𝐸𝐹) ∈ Fin ∧ {𝑥, 𝑦} ⊆ (𝐸𝐹) ∧ {𝑥, 𝑦} ≈ (𝐸𝐹)) → {𝑥, 𝑦} = (𝐸𝐹))
5735, 37, 55, 56syl3anc 1370 . . . . . . . . . . . . . 14 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} = (𝐸𝐹))
5857eqcomd 2742 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝐸𝐹) = {𝑥, 𝑦})
5933, 58jca 512 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝑦𝑉 ∧ (𝐸𝐹) = {𝑥, 𝑦}))
6059expr 457 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → (𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}) → (𝑦𝑉 ∧ (𝐸𝐹) = {𝑥, 𝑦})))
6160eximdv 1919 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → (∃𝑦 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}) → ∃𝑦(𝑦𝑉 ∧ (𝐸𝐹) = {𝑥, 𝑦})))
6261imp 407 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ∃𝑦 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥})) → ∃𝑦(𝑦𝑉 ∧ (𝐸𝐹) = {𝑥, 𝑦}))
63 df-rex 3071 . . . . . . . . 9 (∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦} ↔ ∃𝑦(𝑦𝑉 ∧ (𝐸𝐹) = {𝑥, 𝑦}))
6462, 63sylibr 233 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ∃𝑦 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥})) → ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
6529, 64sylan2b 594 . . . . . . 7 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) ≠ ∅) → ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
6628, 65pm2.61dane 3029 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
6715, 66jca 512 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → (𝑥𝑉 ∧ ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦}))
6867ex 413 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝑥 ∈ (𝐸𝐹) → (𝑥𝑉 ∧ ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})))
6968eximdv 1919 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (∃𝑥 𝑥 ∈ (𝐸𝐹) → ∃𝑥(𝑥𝑉 ∧ ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})))
705, 69mpd 15 . 2 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → ∃𝑥(𝑥𝑉 ∧ ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦}))
71 df-rex 3071 . 2 (∃𝑥𝑉𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦} ↔ ∃𝑥(𝑥𝑉 ∧ ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦}))
7270, 71sylibr 233 1 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → ∃𝑥𝑉𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wex 1780  wcel 2105  wne 2940  wrex 3070  Vcvv 3440  cdif 3893  wss 3896  c0 4266  {csn 4570  {cpr 4572   class class class wbr 5086  dom cdm 5607   Fn wfn 6460  cfv 6465  cen 8779  cdom 8780  Fincfn 8782  cle 11089  2c2 12107  chash 14123  Vtxcvtx 27499  iEdgciedg 27500  UPGraphcupgr 27583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629  ax-cnex 11006  ax-resscn 11007  ax-1cn 11008  ax-icn 11009  ax-addcl 11010  ax-addrcl 11011  ax-mulcl 11012  ax-mulrcl 11013  ax-mulcom 11014  ax-addass 11015  ax-mulass 11016  ax-distr 11017  ax-i2m1 11018  ax-1ne0 11019  ax-1rid 11020  ax-rnegex 11021  ax-rrecex 11022  ax-cnre 11023  ax-pre-lttri 11024  ax-pre-lttrn 11025  ax-pre-ltadd 11026  ax-pre-mulgt0 11027
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-int 4892  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5170  df-tr 5204  df-id 5506  df-eprel 5512  df-po 5520  df-so 5521  df-fr 5562  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-riota 7273  df-ov 7319  df-oprab 7320  df-mpo 7321  df-om 7759  df-1st 7877  df-2nd 7878  df-frecs 8145  df-wrecs 8176  df-recs 8250  df-rdg 8289  df-1o 8345  df-oadd 8349  df-er 8547  df-en 8783  df-dom 8784  df-sdom 8785  df-fin 8786  df-dju 9736  df-card 9774  df-pnf 11090  df-mnf 11091  df-xr 11092  df-ltxr 11093  df-le 11094  df-sub 11286  df-neg 11287  df-nn 12053  df-2 12115  df-n0 12313  df-xnn0 12385  df-z 12399  df-uz 12662  df-fz 13319  df-hash 14124  df-upgr 27585
This theorem is referenced by: (None)
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