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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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