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Theorem omlimcl 7703
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 5826 . . . 4 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 omcl 7661 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
3 eloni 5771 . . . . 5 ((𝐴 ·𝑜 𝐵) ∈ On → Ord (𝐴 ·𝑜 𝐵))
42, 3syl 17 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ·𝑜 𝐵))
51, 4sylan2 490 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Ord (𝐴 ·𝑜 𝐵))
65adantr 480 . 2 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Ord (𝐴 ·𝑜 𝐵))
7 0ellim 5825 . . . . . . . 8 (Lim 𝐵 → ∅ ∈ 𝐵)
8 n0i 3953 . . . . . . . 8 (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
97, 8syl 17 . . . . . . 7 (Lim 𝐵 → ¬ 𝐵 = ∅)
10 n0i 3953 . . . . . . 7 (∅ ∈ 𝐴 → ¬ 𝐴 = ∅)
119, 10anim12ci 590 . . . . . 6 ((Lim 𝐵 ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
1211adantll 750 . . . . 5 (((𝐵𝐶 ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
1312adantll 750 . . . 4 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
14 om00 7700 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))
1514notbid 307 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 ·𝑜 𝐵) = ∅ ↔ ¬ (𝐴 = ∅ ∨ 𝐵 = ∅)))
16 ioran 510 . . . . . . 7 (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
1715, 16syl6bb 276 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 ·𝑜 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
181, 17sylan2 490 . . . . 5 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (¬ (𝐴 ·𝑜 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
1918adantr 480 . . . 4 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (¬ (𝐴 ·𝑜 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
2013, 19mpbird 247 . . 3 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ (𝐴 ·𝑜 𝐵) = ∅)
21 vex 3234 . . . . . . . . . . 11 𝑦 ∈ V
2221sucid 5842 . . . . . . . . . 10 𝑦 ∈ suc 𝑦
23 omlim 7658 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 ·𝑜 𝐵) = 𝑥𝐵 (𝐴 ·𝑜 𝑥))
24 eqeq1 2655 . . . . . . . . . . . 12 ((𝐴 ·𝑜 𝐵) = suc 𝑦 → ((𝐴 ·𝑜 𝐵) = 𝑥𝐵 (𝐴 ·𝑜 𝑥) ↔ suc 𝑦 = 𝑥𝐵 (𝐴 ·𝑜 𝑥)))
2524biimpac 502 . . . . . . . . . . 11 (((𝐴 ·𝑜 𝐵) = 𝑥𝐵 (𝐴 ·𝑜 𝑥) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → suc 𝑦 = 𝑥𝐵 (𝐴 ·𝑜 𝑥))
2623, 25sylan 487 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → suc 𝑦 = 𝑥𝐵 (𝐴 ·𝑜 𝑥))
2722, 26syl5eleq 2736 . . . . . . . . 9 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → 𝑦 𝑥𝐵 (𝐴 ·𝑜 𝑥))
28 eliun 4556 . . . . . . . . 9 (𝑦 𝑥𝐵 (𝐴 ·𝑜 𝑥) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴 ·𝑜 𝑥))
2927, 28sylib 208 . . . . . . . 8 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → ∃𝑥𝐵 𝑦 ∈ (𝐴 ·𝑜 𝑥))
3029adantlr 751 . . . . . . 7 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → ∃𝑥𝐵 𝑦 ∈ (𝐴 ·𝑜 𝑥))
31 onelon 5786 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
321, 31sylan 487 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝑥 ∈ On)
33 onnbtwn 5856 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On → ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
34 imnan 437 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥) ↔ ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
3533, 34sylibr 224 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥))
3635com12 32 . . . . . . . . . . . . . . . 16 (𝑥𝐵 → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3736adantl 481 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3832, 37mpd 15 . . . . . . . . . . . . . 14 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → ¬ 𝐵 ∈ suc 𝑥)
3938adantll 750 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑥𝐵) → ¬ 𝐵 ∈ suc 𝑥)
4039adantlr 751 . . . . . . . . . . . 12 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) → ¬ 𝐵 ∈ suc 𝑥)
4140adantr 480 . . . . . . . . . . 11 (((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 ·𝑜 𝑥)) → ¬ 𝐵 ∈ suc 𝑥)
42 simpl 472 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝐵 ∈ On)
4342, 31jca 553 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ On ∧ 𝑥𝐵) → (𝐵 ∈ On ∧ 𝑥 ∈ On))
441, 43sylan 487 . . . . . . . . . . . . . . . 16 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝐵 ∈ On ∧ 𝑥 ∈ On))
4544anim2i 592 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵)) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)))
4645anassrs 681 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑥𝐵) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)))
47 omcl 7661 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ On)
48 eloni 5771 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ·𝑜 𝑥) ∈ On → Ord (𝐴 ·𝑜 𝑥))
49 ordsucelsuc 7064 . . . . . . . . . . . . . . . . . . . . . . 23 (Ord (𝐴 ·𝑜 𝑥) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 ·𝑜 𝑥)))
5048, 49syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ·𝑜 𝑥) ∈ On → (𝑦 ∈ (𝐴 ·𝑜 𝑥) ↔ suc 𝑦 ∈ suc (𝐴 ·𝑜 𝑥)))
51 oa1suc 7656 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ·𝑜 𝑥) ∈ On → ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) = suc (𝐴 ·𝑜 𝑥))
5251eleq2d 2716 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ·𝑜 𝑥) ∈ On → (suc 𝑦 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ↔ suc 𝑦 ∈ suc (𝐴 ·𝑜 𝑥)))
5350, 52bitr4d 271 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ·𝑜 𝑥) ∈ On → (𝑦 ∈ (𝐴 ·𝑜 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜)))
5447, 53syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜)))
5554adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜)))
56 eloni 5771 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ On → Ord 𝐴)
57 ordgt0ge1 7622 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))
5856, 57syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))
5958adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))
60 1on 7612 . . . . . . . . . . . . . . . . . . . . . . . . 25 1𝑜 ∈ On
61 oaword 7674 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1𝑜 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·𝑜 𝑥) ∈ On) → (1𝑜𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
6260, 61mp3an1 1451 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ On ∧ (𝐴 ·𝑜 𝑥) ∈ On) → (1𝑜𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
6347, 62syldan 486 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (1𝑜𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
6459, 63bitrd 268 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
6564biimpa 500 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
66 omsuc 7651 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
6766adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
6865, 67sseqtr4d 3675 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ (𝐴 ·𝑜 suc 𝑥))
6968sseld 3635 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (suc 𝑦 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) → suc 𝑦 ∈ (𝐴 ·𝑜 suc 𝑥)))
7055, 69sylbid 230 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → suc 𝑦 ∈ (𝐴 ·𝑜 suc 𝑥)))
71 eleq1 2718 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ·𝑜 𝐵) = suc 𝑦 → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) ↔ suc 𝑦 ∈ (𝐴 ·𝑜 suc 𝑥)))
7271biimprd 238 . . . . . . . . . . . . . . . . . 18 ((𝐴 ·𝑜 𝐵) = suc 𝑦 → (suc 𝑦 ∈ (𝐴 ·𝑜 suc 𝑥) → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥)))
7370, 72syl9 77 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = suc 𝑦 → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥))))
7473com23 86 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ((𝐴 ·𝑜 𝐵) = suc 𝑦 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥))))
7574adantlrl 756 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ((𝐴 ·𝑜 𝐵) = suc 𝑦 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥))))
76 sucelon 7059 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
77 omord 7693 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ suc 𝑥 ∧ ∅ ∈ 𝐴) ↔ (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥)))
78 simpl 472 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ suc 𝑥 ∧ ∅ ∈ 𝐴) → 𝐵 ∈ suc 𝑥)
7977, 78syl6bir 244 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) → 𝐵 ∈ suc 𝑥))
8076, 79syl3an2b 1403 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∈ On ∧ 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) → 𝐵 ∈ suc 𝑥))
81803comr 1290 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) → 𝐵 ∈ suc 𝑥))
82813expb 1285 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) → 𝐵 ∈ suc 𝑥))
8382adantr 480 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) → 𝐵 ∈ suc 𝑥))
8475, 83syl6d 75 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ((𝐴 ·𝑜 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥)))
8546, 84sylan 487 . . . . . . . . . . . . 13 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑥𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ((𝐴 ·𝑜 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥)))
8685an32s 863 . . . . . . . . . . . 12 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ((𝐴 ·𝑜 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥)))
8786imp 444 . . . . . . . . . . 11 (((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 ·𝑜 𝑥)) → ((𝐴 ·𝑜 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥))
8841, 87mtod 189 . . . . . . . . . 10 (((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 ·𝑜 𝑥)) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦)
8988exp31 629 . . . . . . . . 9 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝑥𝐵 → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦)))
9089rexlimdv 3059 . . . . . . . 8 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (∃𝑥𝐵 𝑦 ∈ (𝐴 ·𝑜 𝑥) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦))
9190adantr 480 . . . . . . 7 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → (∃𝑥𝐵 𝑦 ∈ (𝐴 ·𝑜 𝑥) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦))
9230, 91mpd 15 . . . . . 6 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦)
9392pm2.01da 457 . . . . 5 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦)
9493adantr 480 . . . 4 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑦 ∈ On) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦)
9594nrexdv 3030 . . 3 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ ∃𝑦 ∈ On (𝐴 ·𝑜 𝐵) = suc 𝑦)
96 ioran 510 . . 3 (¬ ((𝐴 ·𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·𝑜 𝐵) = suc 𝑦) ↔ (¬ (𝐴 ·𝑜 𝐵) = ∅ ∧ ¬ ∃𝑦 ∈ On (𝐴 ·𝑜 𝐵) = suc 𝑦))
9720, 95, 96sylanbrc 699 . 2 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ ((𝐴 ·𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·𝑜 𝐵) = suc 𝑦))
98 dflim3 7089 . 2 (Lim (𝐴 ·𝑜 𝐵) ↔ (Ord (𝐴 ·𝑜 𝐵) ∧ ¬ ((𝐴 ·𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·𝑜 𝐵) = suc 𝑦)))
996, 97, 98sylanbrc 699 1 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  wss 3607  c0 3948   ciun 4552  Ord word 5760  Oncon0 5761  Lim wlim 5762  suc csuc 5763  (class class class)co 6690  1𝑜c1o 7598   +𝑜 coa 7602   ·𝑜 comu 7603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610
This theorem is referenced by:  odi  7704  omass  7705
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