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Theorem oarec 8174
 Description: Recursive definition of ordinal addition. Exercise 25 of [Enderton] p. 240. (Contributed by NM, 26-Dec-2004.) (Revised by Mario Carneiro, 30-May-2015.)
Assertion
Ref Expression
oarec ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem oarec
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7144 . . . 4 (𝑧 = ∅ → (𝐴 +o 𝑧) = (𝐴 +o ∅))
2 mpteq1 5119 . . . . . . . 8 (𝑧 = ∅ → (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ ∅ ↦ (𝐴 +o 𝑥)))
3 mpt0 6463 . . . . . . . 8 (𝑥 ∈ ∅ ↦ (𝐴 +o 𝑥)) = ∅
42, 3eqtrdi 2849 . . . . . . 7 (𝑧 = ∅ → (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ∅)
54rneqd 5773 . . . . . 6 (𝑧 = ∅ → ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ran ∅)
6 rn0 5761 . . . . . 6 ran ∅ = ∅
75, 6eqtrdi 2849 . . . . 5 (𝑧 = ∅ → ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ∅)
87uneq2d 4090 . . . 4 (𝑧 = ∅ → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ∅))
91, 8eqeq12d 2814 . . 3 (𝑧 = ∅ → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o ∅) = (𝐴 ∪ ∅)))
10 oveq2 7144 . . . 4 (𝑧 = 𝑤 → (𝐴 +o 𝑧) = (𝐴 +o 𝑤))
11 mpteq1 5119 . . . . . 6 (𝑧 = 𝑤 → (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥𝑤 ↦ (𝐴 +o 𝑥)))
1211rneqd 5773 . . . . 5 (𝑧 = 𝑤 → ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))
1312uneq2d 4090 . . . 4 (𝑧 = 𝑤 → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))))
1410, 13eqeq12d 2814 . . 3 (𝑧 = 𝑤 → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))))
15 oveq2 7144 . . . 4 (𝑧 = suc 𝑤 → (𝐴 +o 𝑧) = (𝐴 +o suc 𝑤))
16 mpteq1 5119 . . . . . 6 (𝑧 = suc 𝑤 → (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))
1716rneqd 5773 . . . . 5 (𝑧 = suc 𝑤 → ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))
1817uneq2d 4090 . . . 4 (𝑧 = suc 𝑤 → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))))
1915, 18eqeq12d 2814 . . 3 (𝑧 = suc 𝑤 → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))))
20 oveq2 7144 . . . 4 (𝑧 = 𝐵 → (𝐴 +o 𝑧) = (𝐴 +o 𝐵))
21 mpteq1 5119 . . . . . 6 (𝑧 = 𝐵 → (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
2221rneqd 5773 . . . . 5 (𝑧 = 𝐵 → ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))
2322uneq2d 4090 . . . 4 (𝑧 = 𝐵 → (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
2420, 23eqeq12d 2814 . . 3 (𝑧 = 𝐵 → ((𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))) ↔ (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))))
25 oa0 8127 . . . 4 (𝐴 ∈ On → (𝐴 +o ∅) = 𝐴)
26 un0 4298 . . . 4 (𝐴 ∪ ∅) = 𝐴
2725, 26eqtr4di 2851 . . 3 (𝐴 ∈ On → (𝐴 +o ∅) = (𝐴 ∪ ∅))
28 uneq1 4083 . . . . . 6 ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) → ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}) = ((𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)}))
29 unass 4093 . . . . . . 7 ((𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ (ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}))
30 rexun 4117 . . . . . . . . . . 11 (∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +o 𝑥) ↔ (∃𝑥𝑤 𝑦 = (𝐴 +o 𝑥) ∨ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥)))
31 df-suc 6166 . . . . . . . . . . . 12 suc 𝑤 = (𝑤 ∪ {𝑤})
3231rexeqi 3363 . . . . . . . . . . 11 (∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥 ∈ (𝑤 ∪ {𝑤})𝑦 = (𝐴 +o 𝑥))
33 eqid 2798 . . . . . . . . . . . . . 14 (𝑥𝑤 ↦ (𝐴 +o 𝑥)) = (𝑥𝑤 ↦ (𝐴 +o 𝑥))
3433elrnmpt 5793 . . . . . . . . . . . . 13 (𝑦 ∈ V → (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥𝑤 𝑦 = (𝐴 +o 𝑥)))
3534elv 3446 . . . . . . . . . . . 12 (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥𝑤 𝑦 = (𝐴 +o 𝑥))
36 velsn 4541 . . . . . . . . . . . . 13 (𝑦 ∈ {(𝐴 +o 𝑤)} ↔ 𝑦 = (𝐴 +o 𝑤))
37 vex 3444 . . . . . . . . . . . . . 14 𝑤 ∈ V
38 oveq2 7144 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝐴 +o 𝑥) = (𝐴 +o 𝑤))
3938eqeq2d 2809 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → (𝑦 = (𝐴 +o 𝑥) ↔ 𝑦 = (𝐴 +o 𝑤)))
4037, 39rexsn 4580 . . . . . . . . . . . . 13 (∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥) ↔ 𝑦 = (𝐴 +o 𝑤))
4136, 40bitr4i 281 . . . . . . . . . . . 12 (𝑦 ∈ {(𝐴 +o 𝑤)} ↔ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥))
4235, 41orbi12i 912 . . . . . . . . . . 11 ((𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∨ 𝑦 ∈ {(𝐴 +o 𝑤)}) ↔ (∃𝑥𝑤 𝑦 = (𝐴 +o 𝑥) ∨ ∃𝑥 ∈ {𝑤}𝑦 = (𝐴 +o 𝑥)))
4330, 32, 423bitr4i 306 . . . . . . . . . 10 (∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +o 𝑥) ↔ (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∨ 𝑦 ∈ {(𝐴 +o 𝑤)}))
44 eqid 2798 . . . . . . . . . . 11 (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))
45 ovex 7169 . . . . . . . . . . 11 (𝐴 +o 𝑥) ∈ V
4644, 45elrnmpti 5797 . . . . . . . . . 10 (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 ∈ suc 𝑤𝑦 = (𝐴 +o 𝑥))
47 elun 4076 . . . . . . . . . 10 (𝑦 ∈ (ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}) ↔ (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∨ 𝑦 ∈ {(𝐴 +o 𝑤)}))
4843, 46, 473bitr4i 306 . . . . . . . . 9 (𝑦 ∈ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) ↔ 𝑦 ∈ (ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}))
4948eqriv 2795 . . . . . . . 8 ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)) = (ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)})
5049uneq2i 4087 . . . . . . 7 (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ (ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ∪ {(𝐴 +o 𝑤)}))
5129, 50eqtr4i 2824 . . . . . 6 ((𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))
5228, 51eqtrdi 2849 . . . . 5 ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) → ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))))
53 oasuc 8135 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +o suc 𝑤) = suc (𝐴 +o 𝑤))
54 df-suc 6166 . . . . . . 7 suc (𝐴 +o 𝑤) = ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)})
5553, 54eqtrdi 2849 . . . . . 6 ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +o suc 𝑤) = ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}))
5655eqeq1d 2800 . . . . 5 ((𝐴 ∈ On ∧ 𝑤 ∈ On) → ((𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))) ↔ ((𝐴 +o 𝑤) ∪ {(𝐴 +o 𝑤)}) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))))
5752, 56syl5ibr 249 . . . 4 ((𝐴 ∈ On ∧ 𝑤 ∈ On) → ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) → (𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥)))))
5857expcom 417 . . 3 (𝑤 ∈ On → (𝐴 ∈ On → ((𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) → (𝐴 +o suc 𝑤) = (𝐴 ∪ ran (𝑥 ∈ suc 𝑤 ↦ (𝐴 +o 𝑥))))))
59 vex 3444 . . . . . . . 8 𝑧 ∈ V
60 oalim 8143 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝑧 ∈ V ∧ Lim 𝑧)) → (𝐴 +o 𝑧) = 𝑤𝑧 (𝐴 +o 𝑤))
6159, 60mpanr1 702 . . . . . . 7 ((𝐴 ∈ On ∧ Lim 𝑧) → (𝐴 +o 𝑧) = 𝑤𝑧 (𝐴 +o 𝑤))
6261ancoms 462 . . . . . 6 ((Lim 𝑧𝐴 ∈ On) → (𝐴 +o 𝑧) = 𝑤𝑧 (𝐴 +o 𝑤))
6362adantr 484 . . . . 5 (((Lim 𝑧𝐴 ∈ On) ∧ ∀𝑤𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))) → (𝐴 +o 𝑧) = 𝑤𝑧 (𝐴 +o 𝑤))
64 iuneq2 4901 . . . . . 6 (∀𝑤𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) → 𝑤𝑧 (𝐴 +o 𝑤) = 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))))
6564adantl 485 . . . . 5 (((Lim 𝑧𝐴 ∈ On) ∧ ∀𝑤𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))) → 𝑤𝑧 (𝐴 +o 𝑤) = 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))))
66 iunun 4979 . . . . . . 7 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) = ( 𝑤𝑧 𝐴 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))
67 0ellim 6222 . . . . . . . . 9 (Lim 𝑧 → ∅ ∈ 𝑧)
68 ne0i 4250 . . . . . . . . 9 (∅ ∈ 𝑧𝑧 ≠ ∅)
69 iunconst 4891 . . . . . . . . 9 (𝑧 ≠ ∅ → 𝑤𝑧 𝐴 = 𝐴)
7067, 68, 693syl 18 . . . . . . . 8 (Lim 𝑧 𝑤𝑧 𝐴 = 𝐴)
71 df-rex 3112 . . . . . . . . . . . . . 14 (∃𝑥𝑤 𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥(𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
7235, 71bitri 278 . . . . . . . . . . . . 13 (𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥(𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
7372rexbii 3210 . . . . . . . . . . . 12 (∃𝑤𝑧 𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑤𝑧𝑥(𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
74 eluni2 4805 . . . . . . . . . . . . . . . 16 (𝑥 𝑧 ↔ ∃𝑤𝑧 𝑥𝑤)
7574anbi1i 626 . . . . . . . . . . . . . . 15 ((𝑥 𝑧𝑦 = (𝐴 +o 𝑥)) ↔ (∃𝑤𝑧 𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
76 r19.41v 3300 . . . . . . . . . . . . . . 15 (∃𝑤𝑧 (𝑥𝑤𝑦 = (𝐴 +o 𝑥)) ↔ (∃𝑤𝑧 𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
7775, 76bitr4i 281 . . . . . . . . . . . . . 14 ((𝑥 𝑧𝑦 = (𝐴 +o 𝑥)) ↔ ∃𝑤𝑧 (𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
7877exbii 1849 . . . . . . . . . . . . 13 (∃𝑥(𝑥 𝑧𝑦 = (𝐴 +o 𝑥)) ↔ ∃𝑥𝑤𝑧 (𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
79 df-rex 3112 . . . . . . . . . . . . 13 (∃𝑥 𝑧𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥(𝑥 𝑧𝑦 = (𝐴 +o 𝑥)))
80 rexcom4 3212 . . . . . . . . . . . . 13 (∃𝑤𝑧𝑥(𝑥𝑤𝑦 = (𝐴 +o 𝑥)) ↔ ∃𝑥𝑤𝑧 (𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
8178, 79, 803bitr4i 306 . . . . . . . . . . . 12 (∃𝑥 𝑧𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑤𝑧𝑥(𝑥𝑤𝑦 = (𝐴 +o 𝑥)))
8273, 81bitr4i 281 . . . . . . . . . . 11 (∃𝑤𝑧 𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥 𝑧𝑦 = (𝐴 +o 𝑥))
83 limuni 6220 . . . . . . . . . . . 12 (Lim 𝑧𝑧 = 𝑧)
8483rexeqdv 3365 . . . . . . . . . . 11 (Lim 𝑧 → (∃𝑥𝑧 𝑦 = (𝐴 +o 𝑥) ↔ ∃𝑥 𝑧𝑦 = (𝐴 +o 𝑥)))
8582, 84bitr4id 293 . . . . . . . . . 10 (Lim 𝑧 → (∃𝑤𝑧 𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥𝑧 𝑦 = (𝐴 +o 𝑥)))
86 eliun 4886 . . . . . . . . . 10 (𝑦 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑤𝑧 𝑦 ∈ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))
87 eqid 2798 . . . . . . . . . . 11 (𝑥𝑧 ↦ (𝐴 +o 𝑥)) = (𝑥𝑧 ↦ (𝐴 +o 𝑥))
8887, 45elrnmpti 5797 . . . . . . . . . 10 (𝑦 ∈ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)) ↔ ∃𝑥𝑧 𝑦 = (𝐴 +o 𝑥))
8985, 86, 883bitr4g 317 . . . . . . . . 9 (Lim 𝑧 → (𝑦 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) ↔ 𝑦 ∈ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))))
9089eqrdv 2796 . . . . . . . 8 (Lim 𝑧 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)) = ran (𝑥𝑧 ↦ (𝐴 +o 𝑥)))
9170, 90uneq12d 4091 . . . . . . 7 (Lim 𝑧 → ( 𝑤𝑧 𝐴 𝑤𝑧 ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))))
9266, 91syl5eq 2845 . . . . . 6 (Lim 𝑧 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))))
9392ad2antrr 725 . . . . 5 (((Lim 𝑧𝐴 ∈ On) ∧ ∀𝑤𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))) → 𝑤𝑧 (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))))
9463, 65, 933eqtrd 2837 . . . 4 (((Lim 𝑧𝐴 ∈ On) ∧ ∀𝑤𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥)))) → (𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))))
9594exp31 423 . . 3 (Lim 𝑧 → (𝐴 ∈ On → (∀𝑤𝑧 (𝐴 +o 𝑤) = (𝐴 ∪ ran (𝑥𝑤 ↦ (𝐴 +o 𝑥))) → (𝐴 +o 𝑧) = (𝐴 ∪ ran (𝑥𝑧 ↦ (𝐴 +o 𝑥))))))
969, 14, 19, 24, 27, 58, 95tfinds3 7562 . 2 (𝐵 ∈ On → (𝐴 ∈ On → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥)))))
9796impcom 411 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +o 𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ∪ cun 3879  ∅c0 4243  {csn 4525  ∪ cuni 4801  ∪ ciun 4882   ↦ cmpt 5111  ran crn 5521  Oncon0 6160  Lim wlim 6161  suc csuc 6162  (class class class)co 7136   +o coa 8085 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-oadd 8092 This theorem is referenced by:  oacomf1o  8177  onadju  9607
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