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Theorem omlimcl 7518
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 5687 . . . 4 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
2 omcl 7476 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
3 eloni 5632 . . . . 5 ((𝐴 ·𝑜 𝐵) ∈ On → Ord (𝐴 ·𝑜 𝐵))
42, 3syl 17 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ·𝑜 𝐵))
51, 4sylan2 489 . . 3 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Ord (𝐴 ·𝑜 𝐵))
65adantr 479 . 2 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Ord (𝐴 ·𝑜 𝐵))
7 0ellim 5686 . . . . . . . 8 (Lim 𝐵 → ∅ ∈ 𝐵)
8 n0i 3874 . . . . . . . 8 (∅ ∈ 𝐵 → ¬ 𝐵 = ∅)
97, 8syl 17 . . . . . . 7 (Lim 𝐵 → ¬ 𝐵 = ∅)
10 n0i 3874 . . . . . . 7 (∅ ∈ 𝐴 → ¬ 𝐴 = ∅)
119, 10anim12ci 588 . . . . . 6 ((Lim 𝐵 ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
1211adantll 745 . . . . 5 (((𝐵𝐶 ∧ Lim 𝐵) ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
1312adantll 745 . . . 4 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
14 om00 7515 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))
1514notbid 306 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 ·𝑜 𝐵) = ∅ ↔ ¬ (𝐴 = ∅ ∨ 𝐵 = ∅)))
16 ioran 509 . . . . . . 7 (¬ (𝐴 = ∅ ∨ 𝐵 = ∅) ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅))
1715, 16syl6bb 274 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ (𝐴 ·𝑜 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
181, 17sylan2 489 . . . . 5 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (¬ (𝐴 ·𝑜 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
1918adantr 479 . . . 4 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (¬ (𝐴 ·𝑜 𝐵) = ∅ ↔ (¬ 𝐴 = ∅ ∧ ¬ 𝐵 = ∅)))
2013, 19mpbird 245 . . 3 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ (𝐴 ·𝑜 𝐵) = ∅)
21 vex 3171 . . . . . . . . . . 11 𝑦 ∈ V
2221sucid 5703 . . . . . . . . . 10 𝑦 ∈ suc 𝑦
23 omlim 7473 . . . . . . . . . . 11 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐴 ·𝑜 𝐵) = 𝑥𝐵 (𝐴 ·𝑜 𝑥))
24 eqeq1 2609 . . . . . . . . . . . 12 ((𝐴 ·𝑜 𝐵) = suc 𝑦 → ((𝐴 ·𝑜 𝐵) = 𝑥𝐵 (𝐴 ·𝑜 𝑥) ↔ suc 𝑦 = 𝑥𝐵 (𝐴 ·𝑜 𝑥)))
2524biimpac 501 . . . . . . . . . . 11 (((𝐴 ·𝑜 𝐵) = 𝑥𝐵 (𝐴 ·𝑜 𝑥) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → suc 𝑦 = 𝑥𝐵 (𝐴 ·𝑜 𝑥))
2623, 25sylan 486 . . . . . . . . . 10 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → suc 𝑦 = 𝑥𝐵 (𝐴 ·𝑜 𝑥))
2722, 26syl5eleq 2689 . . . . . . . . 9 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → 𝑦 𝑥𝐵 (𝐴 ·𝑜 𝑥))
28 eliun 4450 . . . . . . . . 9 (𝑦 𝑥𝐵 (𝐴 ·𝑜 𝑥) ↔ ∃𝑥𝐵 𝑦 ∈ (𝐴 ·𝑜 𝑥))
2927, 28sylib 206 . . . . . . . 8 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → ∃𝑥𝐵 𝑦 ∈ (𝐴 ·𝑜 𝑥))
3029adantlr 746 . . . . . . 7 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ (𝐴 ·𝑜 𝐵) = suc 𝑦) → ∃𝑥𝐵 𝑦 ∈ (𝐴 ·𝑜 𝑥))
31 onelon 5647 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
321, 31sylan 486 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → 𝑥 ∈ On)
33 onnbtwn 5717 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ On → ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
34 imnan 436 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥) ↔ ¬ (𝑥𝐵𝐵 ∈ suc 𝑥))
3533, 34sylibr 222 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (𝑥𝐵 → ¬ 𝐵 ∈ suc 𝑥))
3635com12 32 . . . . . . . . . . . . . . . 16 (𝑥𝐵 → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3736adantl 480 . . . . . . . . . . . . . . 15 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝑥 ∈ On → ¬ 𝐵 ∈ suc 𝑥))
3832, 37mpd 15 . . . . . . . . . . . . . 14 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → ¬ 𝐵 ∈ suc 𝑥)
3938adantll 745 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑥𝐵) → ¬ 𝐵 ∈ suc 𝑥)
4039adantlr 746 . . . . . . . . . . . 12 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) → ¬ 𝐵 ∈ suc 𝑥)
4140adantr 479 . . . . . . . . . . 11 (((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 ·𝑜 𝑥)) → ¬ 𝐵 ∈ suc 𝑥)
42 simpl 471 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝐵 ∈ On)
4342, 31jca 552 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ On ∧ 𝑥𝐵) → (𝐵 ∈ On ∧ 𝑥 ∈ On))
441, 43sylan 486 . . . . . . . . . . . . . . . 16 (((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵) → (𝐵 ∈ On ∧ 𝑥 ∈ On))
4544anim2i 590 . . . . . . . . . . . . . . 15 ((𝐴 ∈ On ∧ ((𝐵𝐶 ∧ Lim 𝐵) ∧ 𝑥𝐵)) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)))
4645anassrs 677 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑥𝐵) → (𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)))
47 omcl 7476 . . . . . . . . . . . . . . . . . . . . 21 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ On)
48 eloni 5632 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ·𝑜 𝑥) ∈ On → Ord (𝐴 ·𝑜 𝑥))
49 ordsucelsuc 6887 . . . . . . . . . . . . . . . . . . . . . . 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 7471 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ·𝑜 𝑥) ∈ On → ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) = suc (𝐴 ·𝑜 𝑥))
5251eleq2d 2668 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ·𝑜 𝑥) ∈ On → (suc 𝑦 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ↔ suc 𝑦 ∈ suc (𝐴 ·𝑜 𝑥)))
5350, 52bitr4d 269 . . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) ↔ suc 𝑦 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜)))
56 eloni 5632 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴 ∈ On → Ord 𝐴)
57 ordgt0ge1 7437 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))
5856, 57syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐴 ∈ On → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))
5958adantr 479 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ↔ 1𝑜𝐴))
60 1on 7427 . . . . . . . . . . . . . . . . . . . . . . . . 25 1𝑜 ∈ On
61 oaword 7489 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1𝑜 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·𝑜 𝑥) ∈ On) → (1𝑜𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
6260, 61mp3an1 1402 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴 ∈ On ∧ (𝐴 ·𝑜 𝑥) ∈ On) → (1𝑜𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
6347, 62syldan 485 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (1𝑜𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
6459, 63bitrd 266 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
6564biimpa 499 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
66 omsuc 7466 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
6766adantr 479 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
6865, 67sseqtr4d 3600 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) ⊆ (𝐴 ·𝑜 suc 𝑥))
6968sseld 3562 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (suc 𝑦 ∈ ((𝐴 ·𝑜 𝑥) +𝑜 1𝑜) → suc 𝑦 ∈ (𝐴 ·𝑜 suc 𝑥)))
7055, 69sylbid 228 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → suc 𝑦 ∈ (𝐴 ·𝑜 suc 𝑥)))
71 eleq1 2671 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ·𝑜 𝐵) = suc 𝑦 → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) ↔ suc 𝑦 ∈ (𝐴 ·𝑜 suc 𝑥)))
7271biimprd 236 . . . . . . . . . . . . . . . . . 18 ((𝐴 ·𝑜 𝐵) = suc 𝑦 → (suc 𝑦 ∈ (𝐴 ·𝑜 suc 𝑥) → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥)))
7370, 72syl9 74 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) = suc 𝑦 → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥))))
7473com23 83 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ On ∧ 𝑥 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ((𝐴 ·𝑜 𝐵) = suc 𝑦 → (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥))))
7574adantlrl 751 . . . . . . . . . . . . . . 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 6882 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ On ↔ suc 𝑥 ∈ On)
77 omord 7508 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐵 ∈ suc 𝑥 ∧ ∅ ∈ 𝐴) ↔ (𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥)))
78 simpl 471 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ suc 𝑥 ∧ ∅ ∈ 𝐴) → 𝐵 ∈ suc 𝑥)
7977, 78syl6bir 242 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ On ∧ suc 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) → 𝐵 ∈ suc 𝑥))
8076, 79syl3an2b 1354 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∈ On ∧ 𝑥 ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) → 𝐵 ∈ suc 𝑥))
81803comr 1264 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) → 𝐵 ∈ suc 𝑥))
82813expb 1257 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) → 𝐵 ∈ suc 𝑥))
8382adantr 479 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → ((𝐴 ·𝑜 𝐵) ∈ (𝐴 ·𝑜 suc 𝑥) → 𝐵 ∈ suc 𝑥))
8475, 83syl6d 72 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥 ∈ On)) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ((𝐴 ·𝑜 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥)))
8546, 84sylan 486 . . . . . . . . . . . . 13 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ 𝑥𝐵) ∧ ∅ ∈ 𝐴) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ((𝐴 ·𝑜 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥)))
8685an32s 841 . . . . . . . . . . . 12 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ((𝐴 ·𝑜 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥)))
8786imp 443 . . . . . . . . . . 11 (((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 ·𝑜 𝑥)) → ((𝐴 ·𝑜 𝐵) = suc 𝑦𝐵 ∈ suc 𝑥))
8841, 87mtod 187 . . . . . . . . . 10 (((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑥𝐵) ∧ 𝑦 ∈ (𝐴 ·𝑜 𝑥)) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦)
8988exp31 627 . . . . . . . . 9 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝑥𝐵 → (𝑦 ∈ (𝐴 ·𝑜 𝑥) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦)))
9089rexlimdv 3007 . . . . . . . 8 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (∃𝑥𝐵 𝑦 ∈ (𝐴 ·𝑜 𝑥) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦))
9190adantr 479 . . . . . . 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 456 . . . . 5 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦)
9493adantr 479 . . . 4 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝑦 ∈ On) → ¬ (𝐴 ·𝑜 𝐵) = suc 𝑦)
9594nrexdv 2979 . . 3 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ ∃𝑦 ∈ On (𝐴 ·𝑜 𝐵) = suc 𝑦)
96 ioran 509 . . 3 (¬ ((𝐴 ·𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·𝑜 𝐵) = suc 𝑦) ↔ (¬ (𝐴 ·𝑜 𝐵) = ∅ ∧ ¬ ∃𝑦 ∈ On (𝐴 ·𝑜 𝐵) = suc 𝑦))
9720, 95, 96sylanbrc 694 . 2 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → ¬ ((𝐴 ·𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·𝑜 𝐵) = suc 𝑦))
98 dflim3 6912 . 2 (Lim (𝐴 ·𝑜 𝐵) ↔ (Ord (𝐴 ·𝑜 𝐵) ∧ ¬ ((𝐴 ·𝑜 𝐵) = ∅ ∨ ∃𝑦 ∈ On (𝐴 ·𝑜 𝐵) = suc 𝑦)))
996, 97, 98sylanbrc 694 1 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐴 ·𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1975  wrex 2892  wss 3535  c0 3869   ciun 4445  Ord word 5621  Oncon0 5622  Lim wlim 5623  suc csuc 5624  (class class class)co 6523  1𝑜c1o 7413   +𝑜 coa 7417   ·𝑜 comu 7418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
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 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-omul 7425
This theorem is referenced by:  odi  7519  omass  7520
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