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Theorem xpdifid 5745
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 5300 . . . . 5 (𝑝 ∈ ({𝑥} × (𝐵 ∖ {𝑥})) ↔ ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
21rexbii 3188 . . . 4 (∃𝑥𝐴 𝑝 ∈ ({𝑥} × (𝐵 ∖ {𝑥})) ↔ ∃𝑥𝐴𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
3 rexcom4 3378 . . . 4 (∃𝑥𝐴𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ ∃𝑖𝑥𝐴𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
4 rexcom4 3378 . . . . 5 (∃𝑥𝐴𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ ∃𝑗𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
54exbii 1943 . . . 4 (∃𝑖𝑥𝐴𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ ∃𝑖𝑗𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
62, 3, 53bitri 288 . . 3 (∃𝑥𝐴 𝑝 ∈ ({𝑥} × (𝐵 ∖ {𝑥})) ↔ ∃𝑖𝑗𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
7 eliun 4680 . . 3 (𝑝 𝑥𝐴 ({𝑥} × (𝐵 ∖ {𝑥})) ↔ ∃𝑥𝐴 𝑝 ∈ ({𝑥} × (𝐵 ∖ {𝑥})))
8 eldif 3742 . . . . . . 7 (⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I ) ↔ (⟨𝑖, 𝑗⟩ ∈ (𝐴 × 𝐵) ∧ ¬ ⟨𝑖, 𝑗⟩ ∈ I ))
9 opelxp 5313 . . . . . . . 8 (⟨𝑖, 𝑗⟩ ∈ (𝐴 × 𝐵) ↔ (𝑖𝐴𝑗𝐵))
10 df-br 4810 . . . . . . . . . 10 (𝑖 I 𝑗 ↔ ⟨𝑖, 𝑗⟩ ∈ I )
11 vex 3353 . . . . . . . . . . 11 𝑗 ∈ V
1211ideq 5443 . . . . . . . . . 10 (𝑖 I 𝑗𝑖 = 𝑗)
1310, 12bitr3i 268 . . . . . . . . 9 (⟨𝑖, 𝑗⟩ ∈ I ↔ 𝑖 = 𝑗)
1413necon3bbii 2984 . . . . . . . 8 (¬ ⟨𝑖, 𝑗⟩ ∈ I ↔ 𝑖𝑗)
159, 14anbi12i 620 . . . . . . 7 ((⟨𝑖, 𝑗⟩ ∈ (𝐴 × 𝐵) ∧ ¬ ⟨𝑖, 𝑗⟩ ∈ I ) ↔ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
168, 15bitri 266 . . . . . 6 (⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I ) ↔ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
1716anbi2i 616 . . . . 5 ((𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗)))
18172exbii 1944 . . . 4 (∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )) ↔ ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗)))
19 eldifi 3894 . . . . . . . . 9 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → 𝑝 ∈ (𝐴 × 𝐵))
20 elxpi 5299 . . . . . . . . 9 (𝑝 ∈ (𝐴 × 𝐵) → ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖𝐴𝑗𝐵)))
21 simpl 474 . . . . . . . . . 10 ((𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖𝐴𝑗𝐵)) → 𝑝 = ⟨𝑖, 𝑗⟩)
22212eximi 1930 . . . . . . . . 9 (∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖𝐴𝑗𝐵)) → ∃𝑖𝑗 𝑝 = ⟨𝑖, 𝑗⟩)
2319, 20, 223syl 18 . . . . . . . 8 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → ∃𝑖𝑗 𝑝 = ⟨𝑖, 𝑗⟩)
2423ancli 544 . . . . . . 7 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ ∃𝑖𝑗 𝑝 = ⟨𝑖, 𝑗⟩))
25 19.42vv 2050 . . . . . . 7 (∃𝑖𝑗(𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩) ↔ (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ ∃𝑖𝑗 𝑝 = ⟨𝑖, 𝑗⟩))
2624, 25sylibr 225 . . . . . 6 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → ∃𝑖𝑗(𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩))
27 ancom 452 . . . . . . . 8 ((𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ 𝑝 ∈ ((𝐴 × 𝐵) ∖ I )))
28 eleq1 2832 . . . . . . . . . 10 (𝑝 = ⟨𝑖, 𝑗⟩ → (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ↔ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )))
2928adantl 473 . . . . . . . . 9 ((𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩) → (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ↔ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )))
3029pm5.32da 574 . . . . . . . 8 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → ((𝑝 = ⟨𝑖, 𝑗⟩ ∧ 𝑝 ∈ ((𝐴 × 𝐵) ∖ I )) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I ))))
3127, 30syl5bb 274 . . . . . . 7 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → ((𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I ))))
32312exbidv 2019 . . . . . 6 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → (∃𝑖𝑗(𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ∧ 𝑝 = ⟨𝑖, 𝑗⟩) ↔ ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I ))))
3326, 32mpbid 223 . . . . 5 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) → ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )))
3428biimpar 469 . . . . . 6 ((𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )) → 𝑝 ∈ ((𝐴 × 𝐵) ∖ I ))
3534exlimivv 2027 . . . . 5 (∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )) → 𝑝 ∈ ((𝐴 × 𝐵) ∖ I ))
3633, 35impbii 200 . . . 4 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ↔ ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ⟨𝑖, 𝑗⟩ ∈ ((𝐴 × 𝐵) ∖ I )))
37 r19.42v 3239 . . . . . 6 (∃𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
38 simprl 787 . . . . . . . . . . . . 13 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑖 ∈ {𝑦})
39 velsn 4350 . . . . . . . . . . . . 13 (𝑖 ∈ {𝑦} ↔ 𝑖 = 𝑦)
4038, 39sylib 209 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑖 = 𝑦)
41 simpl 474 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑦𝐴)
4240, 41eqeltrd 2844 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑖𝐴)
43 simprr 789 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑗 ∈ (𝐵 ∖ {𝑦}))
4443eldifad 3744 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑗𝐵)
4543eldifbd 3745 . . . . . . . . . . . . . 14 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → ¬ 𝑗 ∈ {𝑦})
46 velsn 4350 . . . . . . . . . . . . . . 15 (𝑗 ∈ {𝑦} ↔ 𝑗 = 𝑦)
4746necon3bbii 2984 . . . . . . . . . . . . . 14 𝑗 ∈ {𝑦} ↔ 𝑗𝑦)
4845, 47sylib 209 . . . . . . . . . . . . 13 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑗𝑦)
4948necomd 2992 . . . . . . . . . . . 12 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑦𝑗)
5040, 49eqnetrd 3004 . . . . . . . . . . 11 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → 𝑖𝑗)
5142, 44, 50jca31 510 . . . . . . . . . 10 ((𝑦𝐴 ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
5251adantll 705 . . . . . . . . 9 (((∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) ∧ 𝑦𝐴) ∧ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))) → ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
53 sneq 4344 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → {𝑥} = {𝑦})
5453eleq2d 2830 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑖 ∈ {𝑥} ↔ 𝑖 ∈ {𝑦}))
5553difeq2d 3890 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑦}))
5655eleq2d 2830 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑗 ∈ (𝐵 ∖ {𝑥}) ↔ 𝑗 ∈ (𝐵 ∖ {𝑦})))
5754, 56anbi12d 624 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) ↔ (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦}))))
5857cbvrexv 3320 . . . . . . . . . 10 (∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) ↔ ∃𝑦𝐴 (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦})))
5958biimpi 207 . . . . . . . . 9 (∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) → ∃𝑦𝐴 (𝑖 ∈ {𝑦} ∧ 𝑗 ∈ (𝐵 ∖ {𝑦})))
6052, 59r19.29a 3225 . . . . . . . 8 (∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) → ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
61 simpll 783 . . . . . . . . 9 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑖𝐴)
62 vsnid 4367 . . . . . . . . . 10 𝑖 ∈ {𝑖}
6362a1i 11 . . . . . . . . 9 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑖 ∈ {𝑖})
64 simplr 785 . . . . . . . . . 10 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑗𝐵)
65 simpr 477 . . . . . . . . . . . 12 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑖𝑗)
6665necomd 2992 . . . . . . . . . . 11 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑗𝑖)
67 velsn 4350 . . . . . . . . . . . 12 (𝑗 ∈ {𝑖} ↔ 𝑗 = 𝑖)
6867necon3bbii 2984 . . . . . . . . . . 11 𝑗 ∈ {𝑖} ↔ 𝑗𝑖)
6966, 68sylibr 225 . . . . . . . . . 10 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → ¬ 𝑗 ∈ {𝑖})
7064, 69eldifd 3743 . . . . . . . . 9 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → 𝑗 ∈ (𝐵 ∖ {𝑖}))
71 sneq 4344 . . . . . . . . . . . 12 (𝑥 = 𝑖 → {𝑥} = {𝑖})
7271eleq2d 2830 . . . . . . . . . . 11 (𝑥 = 𝑖 → (𝑖 ∈ {𝑥} ↔ 𝑖 ∈ {𝑖}))
7371difeq2d 3890 . . . . . . . . . . . 12 (𝑥 = 𝑖 → (𝐵 ∖ {𝑥}) = (𝐵 ∖ {𝑖}))
7473eleq2d 2830 . . . . . . . . . . 11 (𝑥 = 𝑖 → (𝑗 ∈ (𝐵 ∖ {𝑥}) ↔ 𝑗 ∈ (𝐵 ∖ {𝑖})))
7572, 74anbi12d 624 . . . . . . . . . 10 (𝑥 = 𝑖 → ((𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) ↔ (𝑖 ∈ {𝑖} ∧ 𝑗 ∈ (𝐵 ∖ {𝑖}))))
7675rspcev 3461 . . . . . . . . 9 ((𝑖𝐴 ∧ (𝑖 ∈ {𝑖} ∧ 𝑗 ∈ (𝐵 ∖ {𝑖}))) → ∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})))
7761, 63, 70, 76syl12anc 865 . . . . . . . 8 (((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗) → ∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})))
7860, 77impbii 200 . . . . . . 7 (∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥})) ↔ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗))
7978anbi2i 616 . . . . . 6 ((𝑝 = ⟨𝑖, 𝑗⟩ ∧ ∃𝑥𝐴 (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗)))
8037, 79bitri 266 . . . . 5 (∃𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ (𝑝 = ⟨𝑖, 𝑗⟩ ∧ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗)))
81802exbii 1944 . . . 4 (∃𝑖𝑗𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))) ↔ ∃𝑖𝑗(𝑝 = ⟨𝑖, 𝑗⟩ ∧ ((𝑖𝐴𝑗𝐵) ∧ 𝑖𝑗)))
8218, 36, 813bitr4i 294 . . 3 (𝑝 ∈ ((𝐴 × 𝐵) ∖ I ) ↔ ∃𝑖𝑗𝑥𝐴 (𝑝 = ⟨𝑖, 𝑗⟩ ∧ (𝑖 ∈ {𝑥} ∧ 𝑗 ∈ (𝐵 ∖ {𝑥}))))
836, 7, 823bitr4i 294 . 2 (𝑝 𝑥𝐴 ({𝑥} × (𝐵 ∖ {𝑥})) ↔ 𝑝 ∈ ((𝐴 × 𝐵) ∖ I ))
8483eqriv 2762 1 𝑥𝐴 ({𝑥} × (𝐵 ∖ {𝑥})) = ((𝐴 × 𝐵) ∖ I )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wne 2937  wrex 3056  cdif 3729  {csn 4334  cop 4340   ciun 4676   class class class wbr 4809   I cid 5184   × cxp 5275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-iun 4678  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284
This theorem is referenced by:  tglnfn  25733
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