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Theorem upgrex 26595
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 26592 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ≠ ∅)
4 n0 4190 . . . 4 ((𝐸𝐹) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐸𝐹))
53, 4sylib 210 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → ∃𝑥 𝑥 ∈ (𝐸𝐹))
6 simp1 1117 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → 𝐺 ∈ UPGraph)
7 fndm 6285 . . . . . . . . . . . . 13 (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
87eqcomd 2777 . . . . . . . . . . . 12 (𝐸 Fn 𝐴𝐴 = dom 𝐸)
98eleq2d 2844 . . . . . . . . . . 11 (𝐸 Fn 𝐴 → (𝐹𝐴𝐹 ∈ dom 𝐸))
109biimpd 221 . . . . . . . . . 10 (𝐸 Fn 𝐴 → (𝐹𝐴𝐹 ∈ dom 𝐸))
1110a1i 11 . . . . . . . . 9 (𝐺 ∈ UPGraph → (𝐸 Fn 𝐴 → (𝐹𝐴𝐹 ∈ dom 𝐸)))
12113imp 1092 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → 𝐹 ∈ dom 𝐸)
131, 2upgrss 26591 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)
146, 12, 13syl2anc 576 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ⊆ 𝑉)
1514sselda 3851 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → 𝑥𝑉)
1615adantr 473 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → 𝑥𝑉)
17 simpr 477 . . . . . . . . . 10 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → ((𝐸𝐹) ∖ {𝑥}) = ∅)
18 ssdif0 4203 . . . . . . . . . 10 ((𝐸𝐹) ⊆ {𝑥} ↔ ((𝐸𝐹) ∖ {𝑥}) = ∅)
1917, 18sylibr 226 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → (𝐸𝐹) ⊆ {𝑥})
20 simpr 477 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → 𝑥 ∈ (𝐸𝐹))
2120snssd 4612 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → {𝑥} ⊆ (𝐸𝐹))
2221adantr 473 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → {𝑥} ⊆ (𝐸𝐹))
2319, 22eqssd 3868 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → (𝐸𝐹) = {𝑥})
24 preq2 4540 . . . . . . . . . 10 (𝑦 = 𝑥 → {𝑥, 𝑦} = {𝑥, 𝑥})
25 dfsn2 4448 . . . . . . . . . 10 {𝑥} = {𝑥, 𝑥}
2624, 25syl6eqr 2825 . . . . . . . . 9 (𝑦 = 𝑥 → {𝑥, 𝑦} = {𝑥})
2726rspceeqv 3546 . . . . . . . 8 ((𝑥𝑉 ∧ (𝐸𝐹) = {𝑥}) → ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
2816, 23, 27syl2anc 576 . . . . . . 7 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) = ∅) → ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
29 n0 4190 . . . . . . . 8 (((𝐸𝐹) ∖ {𝑥}) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))
3014adantr 473 . . . . . . . . . . . . . 14 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝐸𝐹) ⊆ 𝑉)
31 simprr 761 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))
3231eldifad 3834 . . . . . . . . . . . . . 14 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦 ∈ (𝐸𝐹))
3330, 32sseldd 3852 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦𝑉)
341, 2upgrfi 26594 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ∈ Fin)
3534adantr 473 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝐸𝐹) ∈ Fin)
36 simprl 759 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑥 ∈ (𝐸𝐹))
3736, 32prssd 4625 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} ⊆ (𝐸𝐹))
38 fvex 6509 . . . . . . . . . . . . . . . . 17 (𝐸𝐹) ∈ V
39 ssdomg 8350 . . . . . . . . . . . . . . . . 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 26593 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (♯‘(𝐸𝐹)) ≤ 2)
4241adantr 473 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (♯‘(𝐸𝐹)) ≤ 2)
43 eldifsni 4592 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}) → 𝑦𝑥)
4443ad2antll 717 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑦𝑥)
4544necomd 3015 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → 𝑥𝑦)
46 hashprg 13567 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑦 ↔ (♯‘{𝑥, 𝑦}) = 2))
4746el2v 3415 . . . . . . . . . . . . . . . . . . 19 (𝑥𝑦 ↔ (♯‘{𝑥, 𝑦}) = 2)
4845, 47sylib 210 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (♯‘{𝑥, 𝑦}) = 2)
4942, 48breqtrrd 4953 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (♯‘(𝐸𝐹)) ≤ (♯‘{𝑥, 𝑦}))
50 prfi 8586 . . . . . . . . . . . . . . . . . 18 {𝑥, 𝑦} ∈ Fin
51 hashdom 13551 . . . . . . . . . . . . . . . . . 18 (((𝐸𝐹) ∈ Fin ∧ {𝑥, 𝑦} ∈ Fin) → ((♯‘(𝐸𝐹)) ≤ (♯‘{𝑥, 𝑦}) ↔ (𝐸𝐹) ≼ {𝑥, 𝑦}))
5235, 50, 51sylancl 578 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → ((♯‘(𝐸𝐹)) ≤ (♯‘{𝑥, 𝑦}) ↔ (𝐸𝐹) ≼ {𝑥, 𝑦}))
5349, 52mpbid 224 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝐸𝐹) ≼ {𝑥, 𝑦})
54 sbth 8431 . . . . . . . . . . . . . . . 16 (({𝑥, 𝑦} ≼ (𝐸𝐹) ∧ (𝐸𝐹) ≼ {𝑥, 𝑦}) → {𝑥, 𝑦} ≈ (𝐸𝐹))
5540, 53, 54syl2anc 576 . . . . . . . . . . . . . . 15 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} ≈ (𝐸𝐹))
56 fisseneq 8522 . . . . . . . . . . . . . . 15 (((𝐸𝐹) ∈ Fin ∧ {𝑥, 𝑦} ⊆ (𝐸𝐹) ∧ {𝑥, 𝑦} ≈ (𝐸𝐹)) → {𝑥, 𝑦} = (𝐸𝐹))
5735, 37, 55, 56syl3anc 1352 . . . . . . . . . . . . . 14 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → {𝑥, 𝑦} = (𝐸𝐹))
5857eqcomd 2777 . . . . . . . . . . . . 13 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝐸𝐹) = {𝑥, 𝑦})
5933, 58jca 504 . . . . . . . . . . . 12 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ (𝑥 ∈ (𝐸𝐹) ∧ 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}))) → (𝑦𝑉 ∧ (𝐸𝐹) = {𝑥, 𝑦}))
6059expr 449 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → (𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}) → (𝑦𝑉 ∧ (𝐸𝐹) = {𝑥, 𝑦})))
6160eximdv 1877 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → (∃𝑦 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥}) → ∃𝑦(𝑦𝑉 ∧ (𝐸𝐹) = {𝑥, 𝑦})))
6261imp 398 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ∃𝑦 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥})) → ∃𝑦(𝑦𝑉 ∧ (𝐸𝐹) = {𝑥, 𝑦}))
63 df-rex 3087 . . . . . . . . 9 (∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦} ↔ ∃𝑦(𝑦𝑉 ∧ (𝐸𝐹) = {𝑥, 𝑦}))
6462, 63sylibr 226 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ∃𝑦 𝑦 ∈ ((𝐸𝐹) ∖ {𝑥})) → ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
6529, 64sylan2b 585 . . . . . . 7 ((((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) ∧ ((𝐸𝐹) ∖ {𝑥}) ≠ ∅) → ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
6628, 65pm2.61dane 3048 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
6715, 66jca 504 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) ∧ 𝑥 ∈ (𝐸𝐹)) → (𝑥𝑉 ∧ ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦}))
6867ex 405 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝑥 ∈ (𝐸𝐹) → (𝑥𝑉 ∧ ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})))
6968eximdv 1877 . . 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 3087 . 2 (∃𝑥𝑉𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦} ↔ ∃𝑥(𝑥𝑉 ∧ ∃𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦}))
7270, 71sylibr 226 1 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → ∃𝑥𝑉𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1069   = wceq 1508  wex 1743  wcel 2051  wne 2960  wrex 3082  Vcvv 3408  cdif 3819  wss 3822  c0 4172  {csn 4435  {cpr 4437   class class class wbr 4925  dom cdm 5403   Fn wfn 6180  cfv 6185  cen 8301  cdom 8302  Fincfn 8304  cle 10473  2c2 11493  chash 13503  Vtxcvtx 26499  iEdgciedg 26500  UPGraphcupgr 26583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-nel 3067  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-oadd 7907  df-er 8087  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-dju 9122  df-card 9160  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-nn 11438  df-2 11501  df-n0 11706  df-xnn0 11778  df-z 11792  df-uz 12057  df-fz 12707  df-hash 13504  df-upgr 26585
This theorem is referenced by: (None)
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