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Theorem upgrex 26888
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 26885 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ≠ ∅)
4 n0 4293 . . . 4 ((𝐸𝐹) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐸𝐹))
53, 4sylib 221 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → ∃𝑥 𝑥 ∈ (𝐸𝐹))
6 simp1 1133 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → 𝐺 ∈ UPGraph)
7 fndm 6443 . . . . . . . . . . . . 13 (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
87eqcomd 2830 . . . . . . . . . . . 12 (𝐸 Fn 𝐴𝐴 = dom 𝐸)
98eleq2d 2901 . . . . . . . . . . 11 (𝐸 Fn 𝐴 → (𝐹𝐴𝐹 ∈ dom 𝐸))
109biimpd 232 . . . . . . . . . 10 (𝐸 Fn 𝐴 → (𝐹𝐴𝐹 ∈ dom 𝐸))
1110a1i 11 . . . . . . . . 9 (𝐺 ∈ UPGraph → (𝐸 Fn 𝐴 → (𝐹𝐴𝐹 ∈ dom 𝐸)))
12113imp 1108 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → 𝐹 ∈ dom 𝐸)
131, 2upgrss 26884 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)
146, 12, 13syl2anc 587 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ⊆ 𝑉)
1514sselda 3953 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → 𝑥𝑉)
1615adantr 484 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → 𝑥𝑉)
17 simpr 488 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → ((𝐸𝐹) ∖ {𝑥}) = ∅)
18 ssdif0 4306 . . . . . . . . . 10 ((𝐸𝐹) ⊆ {𝑥} ↔ ((𝐸𝐹) ∖ {𝑥}) = ∅)
1917, 18sylibr 237 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → (𝐸𝐹) ⊆ {𝑥})
20 simpr 488 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → 𝑥 ∈ (𝐸𝐹))
2120snssd 4726 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → {𝑥} ⊆ (𝐸𝐹))
2221adantr 484 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → {𝑥} ⊆ (𝐸𝐹))
2319, 22eqssd 3970 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → (𝐸𝐹) = {𝑥})
24 preq2 4655 . . . . . . . . . 10 (𝑦 = 𝑥 → {𝑥, 𝑦} = {𝑥, 𝑥})
25 dfsn2 4563 . . . . . . . . . 10 {𝑥} = {𝑥, 𝑥}
2624, 25syl6eqr 2877 . . . . . . . . 9 (𝑦 = 𝑥 → {𝑥, 𝑦} = {𝑥})
2726rspceeqv 3624 . . . . . . . 8 ((𝑥𝑉 ∧ (𝐸𝐹) = {𝑥}) → ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
2816, 23, 27syl2anc 587 . . . . . . 7 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
29 n0 4293 . . . . . . . 8 (((𝐸𝐹) ∖ {𝑥}) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))
3014adantr 484 . . . . . . . . . . . . . 14 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝐸𝐹) ⊆ 𝑉)
31 simprr 772 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))
3231eldifad 3931 . . . . . . . . . . . . . 14 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦 ∈ (𝐸𝐹))
3330, 32sseldd 3954 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦𝑉)
341, 2upgrfi 26887 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ∈ Fin)
3534adantr 484 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝐸𝐹) ∈ Fin)
36 simprl 770 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑥 ∈ (𝐸𝐹))
3736, 32prssd 4739 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} ⊆ (𝐸𝐹))
38 fvex 6674 . . . . . . . . . . . . . . . . 17 (𝐸𝐹) ∈ V
39 ssdomg 8551 . . . . . . . . . . . . . . . . 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 26886 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (♯‘(𝐸𝐹)) ≤ 2)
4241adantr 484 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (♯‘(𝐸𝐹)) ≤ 2)
43 eldifsni 4707 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}) → 𝑦𝑥)
4443ad2antll 728 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦𝑥)
4544necomd 3069 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑥𝑦)
46 hashprg 13761 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑦 ↔ (♯‘{𝑥, 𝑦}) = 2))
4746el2v 3487 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑦 ↔ (♯‘{𝑥, 𝑦}) = 2)
4845, 47sylib 221 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (♯‘{𝑥, 𝑦}) = 2)
4942, 48breqtrrd 5080 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (♯‘(𝐸𝐹)) ≤ (♯‘{𝑥, 𝑦}))
50 prfi 8790 . . . . . . . . . . . . . . . . . 18 {𝑥, 𝑦} ∈ Fin
51 hashdom 13745 . . . . . . . . . . . . . . . . . 18 (((𝐸𝐹) ∈ Fin ∧ {𝑥, 𝑦} ∈ Fin) → ((♯‘(𝐸𝐹)) ≤ (♯‘{𝑥, 𝑦}) ↔ (𝐸𝐹) ≼ {𝑥, 𝑦}))
5235, 50, 51sylancl 589 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → ((♯‘(𝐸𝐹)) ≤ (♯‘{𝑥, 𝑦}) ↔ (𝐸𝐹) ≼ {𝑥, 𝑦}))
5349, 52mpbid 235 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝐸𝐹) ≼ {𝑥, 𝑦})
54 sbth 8634 . . . . . . . . . . . . . . . 16 (({𝑥, 𝑦} ≼ (𝐸𝐹) ∧ (𝐸𝐹) ≼ {𝑥, 𝑦}) → {𝑥, 𝑦} ≈ (𝐸𝐹))
5540, 53, 54syl2anc 587 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} ≈ (𝐸𝐹))
56 fisseneq 8726 . . . . . . . . . . . . . . 15 (((𝐸𝐹) ∈ Fin ∧ {𝑥, 𝑦} ⊆ (𝐸𝐹) ∧ {𝑥, 𝑦} ≈ (𝐸𝐹)) → {𝑥, 𝑦} = (𝐸𝐹))
5735, 37, 55, 56syl3anc 1368 . . . . . . . . . . . . . 14 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} = (𝐸𝐹))
5857eqcomd 2830 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝐸𝐹) = {𝑥, 𝑦})
5933, 58jca 515 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝑦𝑉 ∧ (𝐸𝐹) = {𝑥, 𝑦}))
6059expr 460 . . . . . . . . . . 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 410 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ∃𝑦 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥})) → ∃𝑦(𝑦𝑉 ∧ (𝐸𝐹) = {𝑥, 𝑦}))
63 df-rex 3139 . . . . . . . . 9 (∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦} ↔ ∃𝑦(𝑦𝑉 ∧ (𝐸𝐹) = {𝑥, 𝑦}))
6462, 63sylibr 237 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ∃𝑦 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥})) → ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
6529, 64sylan2b 596 . . . . . . 7 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) ≠ ∅) → ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
6628, 65pm2.61dane 3101 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
6715, 66jca 515 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → (𝑥𝑉 ∧ ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦}))
6867ex 416 . . . 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 3139 . 2 (∃𝑥𝑉𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦} ↔ ∃𝑥(𝑥𝑉 ∧ ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦}))
7270, 71sylibr 237 1 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → ∃𝑥𝑉𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  wne 3014  wrex 3134  Vcvv 3480  cdif 3916  wss 3919  c0 4276  {csn 4550  {cpr 4552   class class class wbr 5052  dom cdm 5542   Fn wfn 6338  cfv 6343  cen 8502  cdom 8503  Fincfn 8505  cle 10674  2c2 11689  chash 13695  Vtxcvtx 26792  iEdgciedg 26793  UPGraphcupgr 26876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-dju 9327  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-n0 11895  df-xnn0 11965  df-z 11979  df-uz 12241  df-fz 12895  df-hash 13696  df-upgr 26878
This theorem is referenced by: (None)
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