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Theorem upgrex 28043
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 28040 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ≠ ∅)
4 n0 4306 . . . 4 ((𝐸𝐹) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐸𝐹))
53, 4sylib 217 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → ∃𝑥 𝑥 ∈ (𝐸𝐹))
6 simp1 1136 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → 𝐺 ∈ UPGraph)
7 fndm 6605 . . . . . . . . . . . . 13 (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
87eqcomd 2742 . . . . . . . . . . . 12 (𝐸 Fn 𝐴𝐴 = dom 𝐸)
98eleq2d 2823 . . . . . . . . . . 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 1111 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → 𝐹 ∈ dom 𝐸)
131, 2upgrss 28039 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)
146, 12, 13syl2anc 584 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ⊆ 𝑉)
1514sselda 3944 . . . . . 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 4323 . . . . . . . . . 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 4769 . . . . . . . . . 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 3961 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → (𝐸𝐹) = {𝑥})
24 preq2 4695 . . . . . . . . . 10 (𝑦 = 𝑥 → {𝑥, 𝑦} = {𝑥, 𝑥})
25 dfsn2 4599 . . . . . . . . . 10 {𝑥} = {𝑥, 𝑥}
2624, 25eqtr4di 2794 . . . . . . . . 9 (𝑦 = 𝑥 → {𝑥, 𝑦} = {𝑥})
2726rspceeqv 3595 . . . . . . . 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 4306 . . . . . . . 8 (((𝐸𝐹) ∖ {𝑥}) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))
3014adantr 481 . . . . . . . . . . . . . 14 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝐸𝐹) ⊆ 𝑉)
31 simprr 771 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))
3231eldifad 3922 . . . . . . . . . . . . . 14 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦 ∈ (𝐸𝐹))
3330, 32sseldd 3945 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦𝑉)
341, 2upgrfi 28042 . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑥 ∈ (𝐸𝐹))
3736, 32prssd 4782 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} ⊆ (𝐸𝐹))
38 fvex 6855 . . . . . . . . . . . . . . . . 17 (𝐸𝐹) ∈ V
39 ssdomg 8940 . . . . . . . . . . . . . . . . 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 28041 . . . . . . . . . . . . . . . . . . 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 4750 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}) → 𝑦𝑥)
4443ad2antll 727 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦𝑥)
4544necomd 2999 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑥𝑦)
46 hashprg 14295 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑦 ↔ (♯‘{𝑥, 𝑦}) = 2))
4746el2v 3453 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑦 ↔ (♯‘{𝑥, 𝑦}) = 2)
4845, 47sylib 217 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (♯‘{𝑥, 𝑦}) = 2)
4942, 48breqtrrd 5133 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (♯‘(𝐸𝐹)) ≤ (♯‘{𝑥, 𝑦}))
50 prfi 9266 . . . . . . . . . . . . . . . . . 18 {𝑥, 𝑦} ∈ Fin
51 hashdom 14279 . . . . . . . . . . . . . . . . . 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 9037 . . . . . . . . . . . . . . . 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 9201 . . . . . . . . . . . . . . 15 (((𝐸𝐹) ∈ Fin ∧ {𝑥, 𝑦} ⊆ (𝐸𝐹) ∧ {𝑥, 𝑦} ≈ (𝐸𝐹)) → {𝑥, 𝑦} = (𝐸𝐹))
5735, 37, 55, 56syl3anc 1371 . . . . . . . . . . . . . 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 1920 . . . . . . . . . 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 3074 . . . . . . . . 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 3032 . . . . . 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 1920 . . 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 3074 . 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 1087   = wceq 1541  wex 1781  wcel 2106  wne 2943  wrex 3073  Vcvv 3445  cdif 3907  wss 3910  c0 4282  {csn 4586  {cpr 4588   class class class wbr 5105  dom cdm 5633   Fn wfn 6491  cfv 6496  cen 8880  cdom 8881  Fincfn 8883  cle 11190  2c2 12208  chash 14230  Vtxcvtx 27947  iEdgciedg 27948  UPGraphcupgr 28031
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-upgr 28033
This theorem is referenced by: (None)
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