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Theorem xpdifid 6022
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 5576 . . . . 5 (𝑝 ∈ ({𝑥} × (𝐵 ∖ {𝑥})) ↔ ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
21rexbii 3251 . . . 4 (∃𝑥𝐴 𝑝 ∈ ({𝑥} × (𝐵 ∖ {𝑥})) ↔ ∃𝑥𝐴𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
3 rexcom4 3253 . . . 4 (∃𝑥𝐴𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ ∃𝑖𝑥𝐴𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
4 rexcom4 3253 . . . . 5 (∃𝑥𝐴𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ ∃𝑗𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
54exbii 1841 . . . 4 (∃𝑖𝑥𝐴𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ ∃𝑖𝑗𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
62, 3, 53bitri 298 . . 3 (∃𝑥𝐴 𝑝 ∈ ({𝑥} × (𝐵 ∖ {𝑥})) ↔ ∃𝑖𝑗𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
7 eliun 4920 . . 3 (𝑝 𝑥𝐴 ({𝑥} × (𝐵 ∖ {𝑥})) ↔ ∃𝑥𝐴 𝑝 ∈ ({𝑥} × (𝐵 ∖ {𝑥})))
8 eldif 3949 . . . . . . 7 (⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I ) ↔ (⟨𝑖, 𝑗⟩ ∈ (𝐴 × 𝐵) ∧ ¬ ⟨𝑖, 𝑗⟩ ∈ I ))
9 opelxp 5589 . . . . . . . 8 (⟨𝑖, 𝑗⟩ ∈ (𝐴 × 𝐵) ↔ (𝑖𝐴𝑗𝐵))
10 df-br 5063 . . . . . . . . . 10 (𝑖 I 𝑗 ↔ ⟨𝑖, 𝑗⟩ ∈ I )
11 vex 3502 . . . . . . . . . . 11 𝑗 ∈ V
1211ideq 5721 . . . . . . . . . 10 (𝑖 I 𝑗𝑖 = 𝑗)
1310, 12bitr3i 278 . . . . . . . . 9 (⟨𝑖, 𝑗⟩ ∈ I ↔ 𝑖 = 𝑗)
1413necon3bbii 3067 . . . . . . . 8 (¬ ⟨𝑖, 𝑗⟩ ∈ I ↔ 𝑖𝑗)
159, 14anbi12i 626 . . . . . . 7 ((⟨𝑖, 𝑗⟩ ∈ (𝐴 × 𝐵) ∧ ¬ ⟨𝑖, 𝑗⟩ ∈ I ) ↔ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
168, 15bitri 276 . . . . . 6 (⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I ) ↔ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
1716anbi2i 622 . . . . 5 ((𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗)))
18172exbii 1842 . . . 4 (∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )) ↔ ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗)))
19 eldifi 4106 . . . . . . . . 9 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → 𝑝 ∈ (𝐴 × 𝐵))
20 elxpi 5575 . . . . . . . . 9 (𝑝 ∈ (𝐴 × 𝐵) → ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖𝐴𝑗𝐵)))
21 simpl 483 . . . . . . . . . 10 ((𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖𝐴𝑗𝐵)) → 𝑝 = ⟨𝑖, 𝑗⟩)
22212eximi 1829 . . . . . . . . 9 (∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖𝐴𝑗𝐵)) → ∃𝑖𝑗 𝑝 = ⟨𝑖, 𝑗⟩)
2319, 20, 223syl 18 . . . . . . . 8 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → ∃𝑖𝑗 𝑝 = ⟨𝑖, 𝑗⟩)
2423ancli 549 . . . . . . 7 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ ∃𝑖𝑗 𝑝 = ⟨𝑖, 𝑗⟩))
25 19.42vv 1951 . . . . . . 7 (∃𝑖𝑗(𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩) ↔ (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ ∃𝑖𝑗 𝑝 = ⟨𝑖, 𝑗⟩))
2624, 25sylibr 235 . . . . . 6 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → ∃𝑖𝑗(𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩))
27 ancom 461 . . . . . . . 8 ((𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ 𝑝 ∈ ((𝐴 × 𝐵) ∖ I )))
28 eleq1 2904 . . . . . . . . . 10 (𝑝 = ⟨𝑖, 𝑗⟩ → (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ↔ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )))
2928adantl 482 . . . . . . . . 9 ((𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩) → (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ↔ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )))
3029pm5.32da 579 . . . . . . . 8 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → ((𝑝 = ⟨𝑖, 𝑗⟩ ∧ 𝑝 ∈ ((𝐴 × 𝐵) ∖ I )) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I ))))
3127, 30syl5bb 284 . . . . . . 7 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → ((𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I ))))
32312exbidv 1918 . . . . . 6 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → (∃𝑖𝑗(𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩) ↔ ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I ))))
3326, 32mpbid 233 . . . . 5 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )))
3428biimpar 478 . . . . . 6 ((𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )) → 𝑝 ∈ ((𝐴 × 𝐵) ∖ I ))
3534exlimivv 1926 . . . . 5 (∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )) → 𝑝 ∈ ((𝐴 × 𝐵) ∖ I ))
3633, 35impbii 210 . . . 4 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ↔ ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )))
37 r19.42v 3354 . . . . . 6 (∃𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
38 simprl 767 . . . . . . . . . . . . 13 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑖 ∈ {𝑦})
39 velsn 4579 . . . . . . . . . . . . 13 (𝑖 ∈ {𝑦} ↔ 𝑖 = 𝑦)
4038, 39sylib 219 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑖 = 𝑦)
41 simpl 483 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑦𝐴)
4240, 41eqeltrd 2917 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑖𝐴)
43 simprr 769 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑗 ∈ (𝐵 ∖ {𝑦}))
4443eldifad 3951 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑗𝐵)
4543eldifbd 3952 . . . . . . . . . . . . . 14 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → ¬ 𝑗 ∈ {𝑦})
46 velsn 4579 . . . . . . . . . . . . . . 15 (𝑗 ∈ {𝑦} ↔ 𝑗 = 𝑦)
4746necon3bbii 3067 . . . . . . . . . . . . . 14 𝑗 ∈ {𝑦} ↔ 𝑗𝑦)
4845, 47sylib 219 . . . . . . . . . . . . 13 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑗𝑦)
4948necomd 3075 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑦𝑗)
5040, 49eqnetrd 3087 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑖𝑗)
5142, 44, 50jca31 515 . . . . . . . . . 10 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
5251adantll 710 . . . . . . . . 9 (((∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) ∧ 𝑦𝐴) ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
53 sneq 4573 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → {𝑥} = {𝑦})
5453eleq2d 2902 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑖 ∈ {𝑥} ↔ 𝑖 ∈ {𝑦}))
5553difeq2d 4102 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑦}))
5655eleq2d 2902 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑗 ∈ (𝐵 ∖ {𝑥}) ↔ 𝑗 ∈ (𝐵 ∖ {𝑦})))
5754, 56anbi12d 630 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) ↔ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))))
5857cbvrexvw 3455 . . . . . . . . . 10 (∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) ↔ ∃𝑦𝐴 (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦})))
5958biimpi 217 . . . . . . . . 9 (∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) → ∃𝑦𝐴 (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦})))
6052, 59r19.29a 3293 . . . . . . . 8 (∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) → ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
61 simpll 763 . . . . . . . . 9 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑖𝐴)
62 vsnid 4598 . . . . . . . . . 10 𝑖 ∈ {𝑖}
6362a1i 11 . . . . . . . . 9 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑖 ∈ {𝑖})
64 simplr 765 . . . . . . . . . 10 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑗𝐵)
65 simpr 485 . . . . . . . . . . . 12 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑖𝑗)
6665necomd 3075 . . . . . . . . . . 11 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑗𝑖)
67 velsn 4579 . . . . . . . . . . . 12 (𝑗 ∈ {𝑖} ↔ 𝑗 = 𝑖)
6867necon3bbii 3067 . . . . . . . . . . 11 𝑗 ∈ {𝑖} ↔ 𝑗𝑖)
6966, 68sylibr 235 . . . . . . . . . 10 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → ¬ 𝑗 ∈ {𝑖})
7064, 69eldifd 3950 . . . . . . . . 9 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑗 ∈ (𝐵 ∖ {𝑖}))
71 sneq 4573 . . . . . . . . . . . 12 (𝑥 = 𝑖 → {𝑥} = {𝑖})
7271eleq2d 2902 . . . . . . . . . . 11 (𝑥 = 𝑖 → (𝑖 ∈ {𝑥} ↔ 𝑖 ∈ {𝑖}))
7371difeq2d 4102 . . . . . . . . . . . 12 (𝑥 = 𝑖 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑖}))
7473eleq2d 2902 . . . . . . . . . . 11 (𝑥 = 𝑖 → (𝑗 ∈ (𝐵 ∖ {𝑥}) ↔ 𝑗 ∈ (𝐵 ∖ {𝑖})))
7572, 74anbi12d 630 . . . . . . . . . 10 (𝑥 = 𝑖 → ((𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) ↔ (𝑖 ∈ {𝑖} ∧ 𝑗 ∈ (𝐵 ∖ {𝑖}))))
7675rspcev 3626 . . . . . . . . 9 ((𝑖𝐴 ∧ (𝑖 ∈ {𝑖} ∧ 𝑗 ∈ (𝐵 ∖ {𝑖}))) → ∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})))
7761, 63, 70, 76syl12anc 834 . . . . . . . 8 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → ∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})))
7860, 77impbii 210 . . . . . . 7 (∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) ↔ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
7978anbi2i 622 . . . . . 6 ((𝑝 = ⟨𝑖, 𝑗⟩ ∧ ∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗)))
8037, 79bitri 276 . . . . 5 (∃𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗)))
81802exbii 1842 . . . 4 (∃𝑖𝑗𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗)))
8218, 36, 813bitr4i 304 . . 3 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ↔ ∃𝑖𝑗𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
836, 7, 823bitr4i 304 . 2 (𝑝 𝑥𝐴 ({𝑥} × (𝐵 ∖ {𝑥})) ↔ 𝑝 ∈ ((𝐴 × 𝐵) ∖ I ))
8483eqriv 2822 1 𝑥𝐴 ({𝑥} × (𝐵 ∖ {𝑥})) = ((𝐴 × 𝐵) ∖ I )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  wne 3020  wrex 3143  cdif 3936  {csn 4563  cop 4569   ciun 4916   class class class wbr 5062   I cid 5457   × cxp 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-iun 4918  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560
This theorem is referenced by:  tglnfn  26247
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