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Theorem xpdifid 6124
Description: The set of distinct couples in a Cartesian product. (Contributed by Thierry Arnoux, 25-May-2019.)
Assertion
Ref Expression
xpdifid 𝑥𝐴 ({𝑥} × (𝐵 ∖ {𝑥})) = ((𝐴 × 𝐵) ∖ I )
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem xpdifid
Dummy variables 𝑖 𝑗 𝑝 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp 5660 . . . . 5 (𝑝 ∈ ({𝑥} × (𝐵 ∖ {𝑥})) ↔ ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
21rexbii 3094 . . . 4 (∃𝑥𝐴 𝑝 ∈ ({𝑥} × (𝐵 ∖ {𝑥})) ↔ ∃𝑥𝐴𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
3 rexcom4 3270 . . . 4 (∃𝑥𝐴𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ ∃𝑖𝑥𝐴𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
4 rexcom4 3270 . . . . 5 (∃𝑥𝐴𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ ∃𝑗𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
54exbii 1851 . . . 4 (∃𝑖𝑥𝐴𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ ∃𝑖𝑗𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
62, 3, 53bitri 297 . . 3 (∃𝑥𝐴 𝑝 ∈ ({𝑥} × (𝐵 ∖ {𝑥})) ↔ ∃𝑖𝑗𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
7 eliun 4962 . . 3 (𝑝 𝑥𝐴 ({𝑥} × (𝐵 ∖ {𝑥})) ↔ ∃𝑥𝐴 𝑝 ∈ ({𝑥} × (𝐵 ∖ {𝑥})))
8 eldif 3924 . . . . . . 7 (⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I ) ↔ (⟨𝑖, 𝑗⟩ ∈ (𝐴 × 𝐵) ∧ ¬ ⟨𝑖, 𝑗⟩ ∈ I ))
9 opelxp 5673 . . . . . . . 8 (⟨𝑖, 𝑗⟩ ∈ (𝐴 × 𝐵) ↔ (𝑖𝐴𝑗𝐵))
10 df-br 5110 . . . . . . . . . 10 (𝑖 I 𝑗 ↔ ⟨𝑖, 𝑗⟩ ∈ I )
11 vex 3451 . . . . . . . . . . 11 𝑗 ∈ V
1211ideq 5812 . . . . . . . . . 10 (𝑖 I 𝑗𝑖 = 𝑗)
1310, 12bitr3i 277 . . . . . . . . 9 (⟨𝑖, 𝑗⟩ ∈ I ↔ 𝑖 = 𝑗)
1413necon3bbii 2988 . . . . . . . 8 (¬ ⟨𝑖, 𝑗⟩ ∈ I ↔ 𝑖𝑗)
159, 14anbi12i 628 . . . . . . 7 ((⟨𝑖, 𝑗⟩ ∈ (𝐴 × 𝐵) ∧ ¬ ⟨𝑖, 𝑗⟩ ∈ I ) ↔ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
168, 15bitri 275 . . . . . 6 (⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I ) ↔ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
1716anbi2i 624 . . . . 5 ((𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗)))
18172exbii 1852 . . . 4 (∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )) ↔ ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗)))
19 eldifi 4090 . . . . . . . . 9 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → 𝑝 ∈ (𝐴 × 𝐵))
20 elxpi 5659 . . . . . . . . 9 (𝑝 ∈ (𝐴 × 𝐵) → ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖𝐴𝑗𝐵)))
21 simpl 484 . . . . . . . . . 10 ((𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖𝐴𝑗𝐵)) → 𝑝 = ⟨𝑖, 𝑗⟩)
22212eximi 1839 . . . . . . . . 9 (∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖𝐴𝑗𝐵)) → ∃𝑖𝑗 𝑝 = ⟨𝑖, 𝑗⟩)
2319, 20, 223syl 18 . . . . . . . 8 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → ∃𝑖𝑗 𝑝 = ⟨𝑖, 𝑗⟩)
2423ancli 550 . . . . . . 7 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ ∃𝑖𝑗 𝑝 = ⟨𝑖, 𝑗⟩))
25 19.42vv 1962 . . . . . . 7 (∃𝑖𝑗(𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩) ↔ (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ ∃𝑖𝑗 𝑝 = ⟨𝑖, 𝑗⟩))
2624, 25sylibr 233 . . . . . 6 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → ∃𝑖𝑗(𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩))
27 ancom 462 . . . . . . . 8 ((𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ 𝑝 ∈ ((𝐴 × 𝐵) ∖ I )))
28 eleq1 2822 . . . . . . . . . 10 (𝑝 = ⟨𝑖, 𝑗⟩ → (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ↔ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )))
2928adantl 483 . . . . . . . . 9 ((𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩) → (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ↔ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )))
3029pm5.32da 580 . . . . . . . 8 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → ((𝑝 = ⟨𝑖, 𝑗⟩ ∧ 𝑝 ∈ ((𝐴 × 𝐵) ∖ I )) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I ))))
3127, 30bitrid 283 . . . . . . 7 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → ((𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I ))))
32312exbidv 1928 . . . . . 6 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → (∃𝑖𝑗(𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩) ↔ ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I ))))
3326, 32mpbid 231 . . . . 5 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )))
3428biimpar 479 . . . . . 6 ((𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )) → 𝑝 ∈ ((𝐴 × 𝐵) ∖ I ))
3534exlimivv 1936 . . . . 5 (∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )) → 𝑝 ∈ ((𝐴 × 𝐵) ∖ I ))
3633, 35impbii 208 . . . 4 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ↔ ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )))
37 r19.42v 3184 . . . . . 6 (∃𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
38 simprl 770 . . . . . . . . . . . . 13 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑖 ∈ {𝑦})
39 velsn 4606 . . . . . . . . . . . . 13 (𝑖 ∈ {𝑦} ↔ 𝑖 = 𝑦)
4038, 39sylib 217 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑖 = 𝑦)
41 simpl 484 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑦𝐴)
4240, 41eqeltrd 2834 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑖𝐴)
43 simprr 772 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑗 ∈ (𝐵 ∖ {𝑦}))
4443eldifad 3926 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑗𝐵)
4543eldifbd 3927 . . . . . . . . . . . . . 14 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → ¬ 𝑗 ∈ {𝑦})
46 velsn 4606 . . . . . . . . . . . . . . 15 (𝑗 ∈ {𝑦} ↔ 𝑗 = 𝑦)
4746necon3bbii 2988 . . . . . . . . . . . . . 14 𝑗 ∈ {𝑦} ↔ 𝑗𝑦)
4845, 47sylib 217 . . . . . . . . . . . . 13 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑗𝑦)
4948necomd 2996 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑦𝑗)
5040, 49eqnetrd 3008 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑖𝑗)
5142, 44, 50jca31 516 . . . . . . . . . 10 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
5251adantll 713 . . . . . . . . 9 (((∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) ∧ 𝑦𝐴) ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
53 sneq 4600 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → {𝑥} = {𝑦})
5453eleq2d 2820 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑖 ∈ {𝑥} ↔ 𝑖 ∈ {𝑦}))
5553difeq2d 4086 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑦}))
5655eleq2d 2820 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑗 ∈ (𝐵 ∖ {𝑥}) ↔ 𝑗 ∈ (𝐵 ∖ {𝑦})))
5754, 56anbi12d 632 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) ↔ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))))
5857cbvrexvw 3225 . . . . . . . . . 10 (∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) ↔ ∃𝑦𝐴 (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦})))
5958biimpi 215 . . . . . . . . 9 (∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) → ∃𝑦𝐴 (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦})))
6052, 59r19.29a 3156 . . . . . . . 8 (∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) → ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
61 simpll 766 . . . . . . . . 9 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑖𝐴)
62 vsnid 4627 . . . . . . . . . 10 𝑖 ∈ {𝑖}
6362a1i 11 . . . . . . . . 9 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑖 ∈ {𝑖})
64 simplr 768 . . . . . . . . . 10 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑗𝐵)
65 simpr 486 . . . . . . . . . . . 12 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑖𝑗)
6665necomd 2996 . . . . . . . . . . 11 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑗𝑖)
67 velsn 4606 . . . . . . . . . . . 12 (𝑗 ∈ {𝑖} ↔ 𝑗 = 𝑖)
6867necon3bbii 2988 . . . . . . . . . . 11 𝑗 ∈ {𝑖} ↔ 𝑗𝑖)
6966, 68sylibr 233 . . . . . . . . . 10 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → ¬ 𝑗 ∈ {𝑖})
7064, 69eldifd 3925 . . . . . . . . 9 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑗 ∈ (𝐵 ∖ {𝑖}))
71 sneq 4600 . . . . . . . . . . . 12 (𝑥 = 𝑖 → {𝑥} = {𝑖})
7271eleq2d 2820 . . . . . . . . . . 11 (𝑥 = 𝑖 → (𝑖 ∈ {𝑥} ↔ 𝑖 ∈ {𝑖}))
7371difeq2d 4086 . . . . . . . . . . . 12 (𝑥 = 𝑖 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑖}))
7473eleq2d 2820 . . . . . . . . . . 11 (𝑥 = 𝑖 → (𝑗 ∈ (𝐵 ∖ {𝑥}) ↔ 𝑗 ∈ (𝐵 ∖ {𝑖})))
7572, 74anbi12d 632 . . . . . . . . . 10 (𝑥 = 𝑖 → ((𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) ↔ (𝑖 ∈ {𝑖} ∧ 𝑗 ∈ (𝐵 ∖ {𝑖}))))
7675rspcev 3583 . . . . . . . . 9 ((𝑖𝐴 ∧ (𝑖 ∈ {𝑖} ∧ 𝑗 ∈ (𝐵 ∖ {𝑖}))) → ∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})))
7761, 63, 70, 76syl12anc 836 . . . . . . . 8 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → ∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})))
7860, 77impbii 208 . . . . . . 7 (∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) ↔ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
7978anbi2i 624 . . . . . 6 ((𝑝 = ⟨𝑖, 𝑗⟩ ∧ ∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗)))
8037, 79bitri 275 . . . . 5 (∃𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗)))
81802exbii 1852 . . . 4 (∃𝑖𝑗𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗)))
8218, 36, 813bitr4i 303 . . 3 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ↔ ∃𝑖𝑗𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
836, 7, 823bitr4i 303 . 2 (𝑝 𝑥𝐴 ({𝑥} × (𝐵 ∖ {𝑥})) ↔ 𝑝 ∈ ((𝐴 × 𝐵) ∖ I ))
8483eqriv 2730 1 𝑥𝐴 ({𝑥} × (𝐵 ∖ {𝑥})) = ((𝐴 × 𝐵) ∖ I )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2940  wrex 3070  cdif 3911  {csn 4590  cop 4596   ciun 4958   class class class wbr 5109   I cid 5534   × cxp 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-iun 4960  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644
This theorem is referenced by:  tglnfn  27538
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