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Theorem oeeu 7547
Description: The division algorithm for ordinal exponentiation. (Contributed by Mario Carneiro, 25-May-2015.)
Assertion
Ref Expression
oeeu ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃!𝑤𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑥,𝑦,𝑧

Proof of Theorem oeeu
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2609 . . . . . 6 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} = {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}
21oeeulem 7545 . . . . 5 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ( {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∈ On ∧ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ⊆ 𝐵𝐵 ∈ (𝐴𝑜 suc {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})))
32simp1d 1065 . . . 4 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∈ On)
4 elex 3184 . . . 4 ( {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∈ On → {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∈ V)
53, 4syl 17 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∈ V)
6 fvex 6098 . . . 4 (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∈ V
76a1i 11 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∈ V)
8 fvex 6098 . . . 4 (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∈ V
98a1i 11 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∈ V)
10 eqid 2609 . . . 4 (℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵)) = (℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))
11 eqid 2609 . . . 4 (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) = (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵)))
12 eqid 2609 . . . 4 (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) = (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵)))
131, 10, 11, 12oeeui 7546 . . 3 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ (𝑥 = {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)} ∧ 𝑦 = (1st ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))) ∧ 𝑧 = (2nd ‘(℩𝑑𝑏 ∈ On ∃𝑐 ∈ (𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)})(𝑑 = ⟨𝑏, 𝑐⟩ ∧ (((𝐴𝑜 {𝑎 ∈ On ∣ 𝐵 ∈ (𝐴𝑜 𝑎)}) ·𝑜 𝑏) +𝑜 𝑐) = 𝐵))))))
145, 7, 9, 13euotd 4891 . 2 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃!𝑤𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
15 df-3an 1032 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)))
16 ancom 464 . . . . . . . . . . 11 (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ↔ (𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))))
1715, 16bitri 262 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ↔ (𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))))
1817anbi1i 726 . . . . . . . . 9 (((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ ((𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
1918anbi2i 725 . . . . . . . 8 ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
20 an12 833 . . . . . . . 8 ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ((𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
21 anass 678 . . . . . . . 8 (((𝑧 ∈ (𝐴𝑜 𝑥) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜))) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ (𝑧 ∈ (𝐴𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
2219, 20, 213bitri 284 . . . . . . 7 ((𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ (𝑧 ∈ (𝐴𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
2322exbii 1763 . . . . . 6 (∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑧(𝑧 ∈ (𝐴𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
24 df-rex 2901 . . . . . 6 (∃𝑧 ∈ (𝐴𝑜 𝑥)((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑧(𝑧 ∈ (𝐴𝑜 𝑥) ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))))
25 r19.42v 3072 . . . . . 6 (∃𝑧 ∈ (𝐴𝑜 𝑥)((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ (𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
2623, 24, 253bitr2i 286 . . . . 5 (∃𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
27262exbii 1764 . . . 4 (∃𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑥𝑦((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
28 r2ex 3042 . . . 4 (∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵) ↔ ∃𝑥𝑦((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜)) ∧ ∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)))
2927, 28bitr4i 265 . . 3 (∃𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
3029eubii 2479 . 2 (∃!𝑤𝑥𝑦𝑧(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ (𝐴 ∖ 1𝑜) ∧ 𝑧 ∈ (𝐴𝑜 𝑥)) ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵)) ↔ ∃!𝑤𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
3114, 30sylib 206 1 ((𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐵 ∈ (On ∖ 1𝑜)) → ∃!𝑤𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1𝑜)∃𝑧 ∈ (𝐴𝑜 𝑥)(𝑤 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (((𝐴𝑜 𝑥) ·𝑜 𝑦) +𝑜 𝑧) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  ∃!weu 2457  wrex 2896  {crab 2899  Vcvv 3172  cdif 3536  wss 3539  cop 4130  cotp 4132   cuni 4366   cint 4404  Oncon0 5626  suc csuc 5628  cio 5752  cfv 5790  (class class class)co 6527  1st c1st 7034  2nd c2nd 7035  1𝑜c1o 7417  2𝑜c2o 7418   +𝑜 coa 7421   ·𝑜 comu 7422  𝑜 coe 7423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-ot 4133  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-omul 7429  df-oexp 7430
This theorem is referenced by: (None)
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