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Theorem omlimcl 8202
 Description: The product of any nonzero ordinal with a limit ordinal is a limit ordinal. Proposition 8.24 of [TakeutiZaring] p. 64. (Contributed by NM, 25-Dec-2004.)
Assertion
Ref Expression
omlimcl (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·o 𝐵))

Proof of Theorem omlimcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limelon 6243 . . . 4 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 omcl 8159 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
3 eloni 6190 . . . . 5 ((𝐴 ·o 𝐵) ∈ On → Ord (𝐴 ·o 𝐵))
42, 3syl 17 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ·o 𝐵))
51, 4sylan2 595 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Ord (𝐴 ·o 𝐵))
65adantr 484 . 2 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Ord (𝐴 ·o 𝐵))
7 0ellim 6242 . . . . . . . 8 (Lim 𝐵 → ∅ ∈ 𝐵)
8 n0i 4282 . . . . . . . 8 (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
97, 8syl 17 . . . . . . 7 (Lim 𝐵 → ¬ 𝐵 = ∅)
10 n0i 4282 . . . . . . 7 (∅ ∈ 𝐴 → ¬ 𝐴 = ∅)
119, 10anim12ci 616 . . . . . 6 ((Lim 𝐵 ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
1211adantll 713 . . . . 5 (((𝐵𝐶 ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
1312adantll 713 . . . 4 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
14 om00 8199 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))
1514notbid 321 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 ·o 𝐵) = ∅ ↔ ¬ (𝐴 = ∅ ∨ 𝐵 = ∅)))
16 ioran 981 . . . . . . 7 (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
1715, 16syl6bb 290 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 ·o 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
181, 17sylan2 595 . . . . 5 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (¬ (𝐴 ·o 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
1918adantr 484 . . . 4 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (¬ (𝐴 ·o 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
2013, 19mpbird 260 . . 3 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ (𝐴 ·o 𝐵) = ∅)
21 vex 3483 . . . . . . . . . . 11 𝑦 ∈ V
2221sucid 6259 . . . . . . . . . 10 𝑦 ∈ suc 𝑦
23 omlim 8156 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 ·o 𝐵) = 𝑥𝐵 (𝐴 ·o 𝑥))
24 eqeq1 2828 . . . . . . . . . . . 12 ((𝐴 ·o 𝐵) = suc 𝑦 → ((𝐴 ·o 𝐵) = 𝑥𝐵 (𝐴 ·o 𝑥) ↔ suc 𝑦 = 𝑥𝐵 (𝐴 ·o 𝑥)))
2524biimpac 482 . . . . . . . . . . 11 (((𝐴 ·o 𝐵) = 𝑥𝐵 (𝐴 ·o 𝑥) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → suc 𝑦 = 𝑥𝐵 (𝐴 ·o 𝑥))
2623, 25sylan 583 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → suc 𝑦 = 𝑥𝐵 (𝐴 ·o 𝑥))
2722, 26eleqtrid 2922 . . . . . . . . 9 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → 𝑦 𝑥𝐵 (𝐴 ·o 𝑥))
28 eliun 4909 . . . . . . . . 9 (𝑦 𝑥𝐵 (𝐴 ·o 𝑥) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴 ·o 𝑥))
2927, 28sylib 221 . . . . . . . 8 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → ∃𝑥𝐵 𝑦 ∈ (𝐴 ·o 𝑥))
3029adantlr 714 . . . . . . 7 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → ∃𝑥𝐵 𝑦 ∈ (𝐴 ·o 𝑥))
31 onelon 6205 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
321, 31sylan 583 . . . . . . . . . . . 12 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝑥 ∈ On)
33 onnbtwn 6271 . . . . . . . . . . . . . . 15 (𝑥 ∈ On → ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
34 imnan 403 . . . . . . . . . . . . . . 15 ((𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥) ↔ ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
3533, 34sylibr 237 . . . . . . . . . . . . . 14 (𝑥 ∈ On → (𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥))
3635com12 32 . . . . . . . . . . . . 13 (𝑥𝐵 → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3736adantl 485 . . . . . . . . . . . 12 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3832, 37mpd 15 . . . . . . . . . . 11 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → ¬ 𝐵 ∈ suc 𝑥)
3938ad5ant24 760 . . . . . . . . . 10 (((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 ·o 𝑥)) → ¬ 𝐵 ∈ suc 𝑥)
40 simpl 486 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝐵 ∈ On)
4140, 31jca 515 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ 𝑥𝐵) → (𝐵 ∈ On ∧ 𝑥 ∈ On))
421, 41sylan 583 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝐵 ∈ On ∧ 𝑥 ∈ On))
4342anim2i 619 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵)) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)))
4443anassrs 471 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑥𝐵) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)))
45 omcl 8159 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
46 eloni 6190 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ·o 𝑥) ∈ On → Ord (𝐴 ·o 𝑥))
47 ordsucelsuc 7533 . . . . . . . . . . . . . . . . . . . . . 22 (Ord (𝐴 ·o 𝑥) → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 ·o 𝑥)))
4846, 47syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ·o 𝑥) ∈ On → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 ·o 𝑥)))
49 oa1suc 8154 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ·o 𝑥) ∈ On → ((𝐴 ·o 𝑥) +o 1o) = suc (𝐴 ·o 𝑥))
5049eleq2d 2901 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ·o 𝑥) ∈ On → (suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o) ↔ suc 𝑦 ∈ suc (𝐴 ·o 𝑥)))
5148, 50bitr4d 285 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ·o 𝑥) ∈ On → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o)))
5245, 51syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o)))
5352adantr 484 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o)))
54 eloni 6190 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ On → Ord 𝐴)
55 ordgt0ge1 8120 . . . . . . . . . . . . . . . . . . . . . . . 24 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1o𝐴))
5654, 55syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1o𝐴))
5756adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ↔ 1o𝐴))
58 1on 8107 . . . . . . . . . . . . . . . . . . . . . . . 24 1o ∈ On
59 oaword 8173 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1o ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·o 𝑥) ∈ On) → (1o𝐴 ↔ ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴)))
6058, 59mp3an1 1445 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ On ∧ (𝐴 ·o 𝑥) ∈ On) → (1o𝐴 ↔ ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴)))
6145, 60syldan 594 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (1o𝐴 ↔ ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴)))
6257, 61bitrd 282 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ↔ ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴)))
6362biimpa 480 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝑥) +o 1o) ⊆ ((𝐴 ·o 𝑥) +o 𝐴))
64 omsuc 8149 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
6564adantr 484 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
6663, 65sseqtrrd 3994 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝑥) +o 1o) ⊆ (𝐴 ·o suc 𝑥))
6766sseld 3952 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (suc 𝑦 ∈ ((𝐴 ·o 𝑥) +o 1o) → suc 𝑦 ∈ (𝐴 ·o suc 𝑥)))
6853, 67sylbid 243 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) → suc 𝑦 ∈ (𝐴 ·o suc 𝑥)))
69 eleq1 2903 . . . . . . . . . . . . . . . . . 18 ((𝐴 ·o 𝐵) = suc 𝑦 → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) ↔ suc 𝑦 ∈ (𝐴 ·o suc 𝑥)))
7069biimprd 251 . . . . . . . . . . . . . . . . 17 ((𝐴 ·o 𝐵) = suc 𝑦 → (suc 𝑦 ∈ (𝐴 ·o suc 𝑥) → (𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥)))
7168, 70syl9 77 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = suc 𝑦 → (𝑦 ∈ (𝐴 ·o 𝑥) → (𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥))))
7271com23 86 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) → ((𝐴 ·o 𝐵) = suc 𝑦 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥))))
7372adantlrl 719 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) → ((𝐴 ·o 𝐵) = suc 𝑦 → (𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥))))
74 sucelon 7528 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
75 omord 8192 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ suc 𝑥 ∧ ∅ ∈ 𝐴) ↔ (𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥)))
76 simpl 486 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ suc 𝑥 ∧ ∅ ∈ 𝐴) → 𝐵 ∈ suc 𝑥)
7775, 76syl6bir 257 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
7874, 77syl3an2b 1401 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ On ∧ 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
79783comr 1122 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
80793expb 1117 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
8180adantr 484 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) ∈ (𝐴 ·o suc 𝑥) → 𝐵 ∈ suc 𝑥))
8273, 81syl6d 75 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) → ((𝐴 ·o 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥)))
8344, 82sylan 583 . . . . . . . . . . . 12 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑥𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·o 𝑥) → ((𝐴 ·o 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥)))
8483an32s 651 . . . . . . . . . . 11 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) → (𝑦 ∈ (𝐴 ·o 𝑥) → ((𝐴 ·o 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥)))
8584imp 410 . . . . . . . . . 10 (((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 ·o 𝑥)) → ((𝐴 ·o 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥))
8639, 85mtod 201 . . . . . . . . 9 (((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 ·o 𝑥)) → ¬ (𝐴 ·o 𝐵) = suc 𝑦)
8786rexlimdva2 3279 . . . . . . . 8 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (∃𝑥𝐵 𝑦 ∈ (𝐴 ·o 𝑥) → ¬ (𝐴 ·o 𝐵) = suc 𝑦))
8887adantr 484 . . . . . . 7 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → (∃𝑥𝐵 𝑦 ∈ (𝐴 ·o 𝑥) → ¬ (𝐴 ·o 𝐵) = suc 𝑦))
8930, 88mpd 15 . . . . . 6 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·o 𝐵) = suc 𝑦) → ¬ (𝐴 ·o 𝐵) = suc 𝑦)
9089pm2.01da 798 . . . . 5 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ (𝐴 ·o 𝐵) = suc 𝑦)
9190adantr 484 . . . 4 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑦 ∈ On) → ¬ (𝐴 ·o 𝐵) = suc 𝑦)
9291nrexdv 3262 . . 3 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ ∃𝑦 ∈ On (𝐴 ·o 𝐵) = suc 𝑦)
93 ioran 981 . . 3 (¬ ((𝐴 ·o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·o 𝐵) = suc 𝑦) ↔ (¬ (𝐴 ·o 𝐵) = ∅ ∧ ¬ ∃𝑦 ∈ On (𝐴 ·o 𝐵) = suc 𝑦))
9420, 92, 93sylanbrc 586 . 2 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ ((𝐴 ·o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·o 𝐵) = suc 𝑦))
95 dflim3 7558 . 2 (Lim (𝐴 ·o 𝐵) ↔ (Ord (𝐴 ·o 𝐵) ∧ ¬ ((𝐴 ·o 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·o 𝐵) = suc 𝑦)))
966, 94, 95sylanbrc 586 1 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·o 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∃wrex 3134   ⊆ wss 3919  ∅c0 4276  ∪ ciun 4905  Ord word 6179  Oncon0 6180  Lim wlim 6181  suc csuc 6182  (class class class)co 7151  1oc1o 8093   +o coa 8097   ·o comu 8098 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-oadd 8104  df-omul 8105 This theorem is referenced by:  odi  8203  omass  8204
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