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Theorem dflim5 43319
Description: A limit ordinal is either the proper class of ordinals or some nonzero product with omega. (Contributed by RP, 8-Jan-2025.)
Assertion
Ref Expression
dflim5 (Lim 𝐴 ↔ (𝐴 = On ∨ ∃𝑥 ∈ (On ∖ 1o)𝐴 = (ω ·o 𝑥)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem dflim5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 limord 6446 . . . . 5 (Lim 𝐴 → Ord 𝐴)
2 ordeleqon 7801 . . . . . . 7 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
32biimpi 216 . . . . . 6 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
43orcomd 871 . . . . 5 (Ord 𝐴 → (𝐴 = On ∨ 𝐴 ∈ On))
51, 4syl 17 . . . 4 (Lim 𝐴 → (𝐴 = On ∨ 𝐴 ∈ On))
65pm4.71ri 560 . . 3 (Lim 𝐴 ↔ ((𝐴 = On ∨ 𝐴 ∈ On) ∧ Lim 𝐴))
7 andir 1010 . . 3 (((𝐴 = On ∨ 𝐴 ∈ On) ∧ Lim 𝐴) ↔ ((𝐴 = On ∧ Lim 𝐴) ∨ (𝐴 ∈ On ∧ Lim 𝐴)))
86, 7bitri 275 . 2 (Lim 𝐴 ↔ ((𝐴 = On ∧ Lim 𝐴) ∨ (𝐴 ∈ On ∧ Lim 𝐴)))
9 limon 7856 . . . . 5 Lim On
10 limeq 6398 . . . . 5 (𝐴 = On → (Lim 𝐴 ↔ Lim On))
119, 10mpbiri 258 . . . 4 (𝐴 = On → Lim 𝐴)
1211pm4.71i 559 . . 3 (𝐴 = On ↔ (𝐴 = On ∧ Lim 𝐴))
1312orbi1i 913 . 2 ((𝐴 = On ∨ (𝐴 ∈ On ∧ Lim 𝐴)) ↔ ((𝐴 = On ∧ Lim 𝐴) ∨ (𝐴 ∈ On ∧ Lim 𝐴)))
14 simpl 482 . . . . . 6 ((𝐴 ∈ On ∧ Lim 𝐴) → 𝐴 ∈ On)
15 omelon 9684 . . . . . . . 8 ω ∈ On
1615a1i 11 . . . . . . 7 (𝐴 ∈ On → ω ∈ On)
17 id 22 . . . . . . 7 (𝐴 ∈ On → 𝐴 ∈ On)
18 peano1 7911 . . . . . . . . 9 ∅ ∈ ω
1918ne0ii 4350 . . . . . . . 8 ω ≠ ∅
2019a1i 11 . . . . . . 7 (𝐴 ∈ On → ω ≠ ∅)
2116, 17, 203jca 1127 . . . . . 6 (𝐴 ∈ On → (ω ∈ On ∧ 𝐴 ∈ On ∧ ω ≠ ∅))
22 omeulem1 8619 . . . . . 6 ((ω ∈ On ∧ 𝐴 ∈ On ∧ ω ≠ ∅) → ∃𝑥 ∈ On ∃𝑦 ∈ ω ((ω ·o 𝑥) +o 𝑦) = 𝐴)
2314, 21, 223syl 18 . . . . 5 ((𝐴 ∈ On ∧ Lim 𝐴) → ∃𝑥 ∈ On ∃𝑦 ∈ ω ((ω ·o 𝑥) +o 𝑦) = 𝐴)
24 limeq 6398 . . . . . . . . . . . . . . . 16 (((ω ·o 𝑥) +o 𝑦) = 𝐴 → (Lim ((ω ·o 𝑥) +o 𝑦) ↔ Lim 𝐴))
2524biimprd 248 . . . . . . . . . . . . . . 15 (((ω ·o 𝑥) +o 𝑦) = 𝐴 → (Lim 𝐴 → Lim ((ω ·o 𝑥) +o 𝑦)))
26 simplr 769 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → 𝑦 ∈ ω)
27 nnlim 7901 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ω → ¬ Lim 𝑦)
2826, 27syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → ¬ Lim 𝑦)
29 on0eln0 6442 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ On → (∅ ∈ 𝑥𝑥 ≠ ∅))
3029biimprd 248 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ On → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
3130necon1bd 2956 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ On → (¬ ∅ ∈ 𝑥𝑥 = ∅))
3231adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (¬ ∅ ∈ 𝑥𝑥 = ∅))
3332imp 406 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → 𝑥 = ∅)
3433, 26jca 511 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → (𝑥 = ∅ ∧ 𝑦 ∈ ω))
35 simpl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → 𝑥 = ∅)
3635oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (ω ·o 𝑥) = (ω ·o ∅))
37 om0 8554 . . . . . . . . . . . . . . . . . . . . . . . . 25 (ω ∈ On → (ω ·o ∅) = ∅)
3815, 37mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (ω ·o ∅) = ∅)
3936, 38eqtrd 2775 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (ω ·o 𝑥) = ∅)
4039oveq1d 7446 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → ((ω ·o 𝑥) +o 𝑦) = (∅ +o 𝑦))
41 nna0r 8646 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ ω → (∅ +o 𝑦) = 𝑦)
4241adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (∅ +o 𝑦) = 𝑦)
4340, 42eqtrd 2775 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → ((ω ·o 𝑥) +o 𝑦) = 𝑦)
44 limeq 6398 . . . . . . . . . . . . . . . . . . . . 21 (((ω ·o 𝑥) +o 𝑦) = 𝑦 → (Lim ((ω ·o 𝑥) +o 𝑦) ↔ Lim 𝑦))
4534, 43, 443syl 18 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → (Lim ((ω ·o 𝑥) +o 𝑦) ↔ Lim 𝑦))
4628, 45mtbird 325 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → ¬ Lim ((ω ·o 𝑥) +o 𝑦))
4746ex 412 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (¬ ∅ ∈ 𝑥 → ¬ Lim ((ω ·o 𝑥) +o 𝑦)))
48 ovex 7464 . . . . . . . . . . . . . . . . . . . . 21 ((ω ·o 𝑥) +o 𝑦) ∈ V
49 nlimsucg 7863 . . . . . . . . . . . . . . . . . . . . 21 (((ω ·o 𝑥) +o 𝑦) ∈ V → ¬ Lim suc ((ω ·o 𝑥) +o 𝑦))
5048, 49mp1i 13 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ¬ Lim suc ((ω ·o 𝑥) +o 𝑦))
51 nnord 7895 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ω → Ord 𝑦)
52 orduniorsuc 7850 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Ord 𝑦 → (𝑦 = 𝑦𝑦 = suc 𝑦))
5351, 52syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ ω → (𝑦 = 𝑦𝑦 = suc 𝑦))
54 3ianor 1106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (¬ (Ord 𝑦𝑦 ≠ ∅ ∧ 𝑦 = 𝑦) ↔ (¬ Ord 𝑦 ∨ ¬ 𝑦 ≠ ∅ ∨ ¬ 𝑦 = 𝑦))
55 df-lim 6391 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (Lim 𝑦 ↔ (Ord 𝑦𝑦 ≠ ∅ ∧ 𝑦 = 𝑦))
5654, 55xchnxbir 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ Lim 𝑦 ↔ (¬ Ord 𝑦 ∨ ¬ 𝑦 ≠ ∅ ∨ ¬ 𝑦 = 𝑦))
5727, 56sylib 218 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ω → (¬ Ord 𝑦 ∨ ¬ 𝑦 ≠ ∅ ∨ ¬ 𝑦 = 𝑦))
5851pm2.24d 151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ ω → (¬ Ord 𝑦 → (𝑦 = 𝑦𝑦 = ∅)))
59 nne 2942 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑦 ≠ ∅ ↔ 𝑦 = ∅)
6059biimpi 216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑦 ≠ ∅ → 𝑦 = ∅)
6160a1i13 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ ω → (¬ 𝑦 ≠ ∅ → (𝑦 = 𝑦𝑦 = ∅)))
62 pm2.21 123 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 𝑦 = 𝑦 → (𝑦 = 𝑦𝑦 = ∅))
6362a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ ω → (¬ 𝑦 = 𝑦 → (𝑦 = 𝑦𝑦 = ∅)))
6458, 61, 633jaod 1428 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ω → ((¬ Ord 𝑦 ∨ ¬ 𝑦 ≠ ∅ ∨ ¬ 𝑦 = 𝑦) → (𝑦 = 𝑦𝑦 = ∅)))
6557, 64mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ω → (𝑦 = 𝑦𝑦 = ∅))
6665orim1d 967 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ ω → ((𝑦 = 𝑦𝑦 = suc 𝑦) → (𝑦 = ∅ ∨ 𝑦 = suc 𝑦)))
6753, 66mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ ω → (𝑦 = ∅ ∨ 𝑦 = suc 𝑦))
6867ord 864 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ω → (¬ 𝑦 = ∅ → 𝑦 = suc 𝑦))
6968adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (¬ 𝑦 = ∅ → 𝑦 = suc 𝑦))
7069imp 406 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → 𝑦 = suc 𝑦)
7170oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ((ω ·o 𝑥) +o 𝑦) = ((ω ·o 𝑥) +o suc 𝑦))
72 simpl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → 𝑥 ∈ On)
7372adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → 𝑥 ∈ On)
74 omcl 8573 . . . . . . . . . . . . . . . . . . . . . . . 24 ((ω ∈ On ∧ 𝑥 ∈ On) → (ω ·o 𝑥) ∈ On)
7515, 73, 74sylancr 587 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → (ω ·o 𝑥) ∈ On)
76 nnon 7893 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ ω → 𝑦 ∈ On)
77 onuni 7808 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ On → 𝑦 ∈ On)
7876, 77syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ω → 𝑦 ∈ On)
7978adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → 𝑦 ∈ On)
8079adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → 𝑦 ∈ On)
81 oasuc 8561 . . . . . . . . . . . . . . . . . . . . . . 23 (((ω ·o 𝑥) ∈ On ∧ 𝑦 ∈ On) → ((ω ·o 𝑥) +o suc 𝑦) = suc ((ω ·o 𝑥) +o 𝑦))
8275, 80, 81syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ((ω ·o 𝑥) +o suc 𝑦) = suc ((ω ·o 𝑥) +o 𝑦))
8371, 82eqtrd 2775 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ((ω ·o 𝑥) +o 𝑦) = suc ((ω ·o 𝑥) +o 𝑦))
84 limeq 6398 . . . . . . . . . . . . . . . . . . . . 21 (((ω ·o 𝑥) +o 𝑦) = suc ((ω ·o 𝑥) +o 𝑦) → (Lim ((ω ·o 𝑥) +o 𝑦) ↔ Lim suc ((ω ·o 𝑥) +o 𝑦)))
8583, 84syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → (Lim ((ω ·o 𝑥) +o 𝑦) ↔ Lim suc ((ω ·o 𝑥) +o 𝑦)))
8650, 85mtbird 325 . . . . . . . . . . . . . . . . . . 19 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ¬ Lim ((ω ·o 𝑥) +o 𝑦))
8786ex 412 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (¬ 𝑦 = ∅ → ¬ Lim ((ω ·o 𝑥) +o 𝑦)))
8847, 87jaod 859 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → ((¬ ∅ ∈ 𝑥 ∨ ¬ 𝑦 = ∅) → ¬ Lim ((ω ·o 𝑥) +o 𝑦)))
8988con2d 134 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (Lim ((ω ·o 𝑥) +o 𝑦) → ¬ (¬ ∅ ∈ 𝑥 ∨ ¬ 𝑦 = ∅)))
90 anor 984 . . . . . . . . . . . . . . . 16 ((∅ ∈ 𝑥𝑦 = ∅) ↔ ¬ (¬ ∅ ∈ 𝑥 ∨ ¬ 𝑦 = ∅))
9189, 90imbitrrdi 252 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (Lim ((ω ·o 𝑥) +o 𝑦) → (∅ ∈ 𝑥𝑦 = ∅)))
9225, 91syl9 77 . . . . . . . . . . . . . 14 (((ω ·o 𝑥) +o 𝑦) = 𝐴 → ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (Lim 𝐴 → (∅ ∈ 𝑥𝑦 = ∅))))
9392com13 88 . . . . . . . . . . . . 13 (Lim 𝐴 → ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (((ω ·o 𝑥) +o 𝑦) = 𝐴 → (∅ ∈ 𝑥𝑦 = ∅))))
9493adantl 481 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ Lim 𝐴) → ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (((ω ·o 𝑥) +o 𝑦) = 𝐴 → (∅ ∈ 𝑥𝑦 = ∅))))
95943imp 1110 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·o 𝑥) +o 𝑦) = 𝐴) → (∅ ∈ 𝑥𝑦 = ∅))
96 simp2 1136 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·o 𝑥) +o 𝑦) = 𝐴) → (𝑥 ∈ On ∧ 𝑦 ∈ ω))
9796, 72syl 17 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·o 𝑥) +o 𝑦) = 𝐴) → 𝑥 ∈ On)
98 simpl 482 . . . . . . . . . . . . . 14 ((∅ ∈ 𝑥𝑦 = ∅) → ∅ ∈ 𝑥)
9997, 98anim12i 613 . . . . . . . . . . . . 13 ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·o 𝑥) +o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥𝑦 = ∅)) → (𝑥 ∈ On ∧ ∅ ∈ 𝑥))
100 ondif1 8538 . . . . . . . . . . . . 13 (𝑥 ∈ (On ∖ 1o) ↔ (𝑥 ∈ On ∧ ∅ ∈ 𝑥))
10199, 100sylibr 234 . . . . . . . . . . . 12 ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·o 𝑥) +o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥𝑦 = ∅)) → 𝑥 ∈ (On ∖ 1o))
102 simpr 484 . . . . . . . . . . . . . . 15 ((∅ ∈ 𝑥𝑦 = ∅) → 𝑦 = ∅)
103102oveq2d 7447 . . . . . . . . . . . . . 14 ((∅ ∈ 𝑥𝑦 = ∅) → ((ω ·o 𝑥) +o 𝑦) = ((ω ·o 𝑥) +o ∅))
104103adantl 481 . . . . . . . . . . . . 13 ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·o 𝑥) +o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥𝑦 = ∅)) → ((ω ·o 𝑥) +o 𝑦) = ((ω ·o 𝑥) +o ∅))
105 simpl3 1192 . . . . . . . . . . . . 13 ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·o 𝑥) +o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥𝑦 = ∅)) → ((ω ·o 𝑥) +o 𝑦) = 𝐴)
10615, 72, 74sylancr 587 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (ω ·o 𝑥) ∈ On)
107 oa0 8553 . . . . . . . . . . . . . . 15 ((ω ·o 𝑥) ∈ On → ((ω ·o 𝑥) +o ∅) = (ω ·o 𝑥))
10896, 106, 1073syl 18 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·o 𝑥) +o 𝑦) = 𝐴) → ((ω ·o 𝑥) +o ∅) = (ω ·o 𝑥))
109108adantr 480 . . . . . . . . . . . . 13 ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·o 𝑥) +o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥𝑦 = ∅)) → ((ω ·o 𝑥) +o ∅) = (ω ·o 𝑥))
110104, 105, 1093eqtr3d 2783 . . . . . . . . . . . 12 ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·o 𝑥) +o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥𝑦 = ∅)) → 𝐴 = (ω ·o 𝑥))
111101, 110jca 511 . . . . . . . . . . 11 ((((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·o 𝑥) +o 𝑦) = 𝐴) ∧ (∅ ∈ 𝑥𝑦 = ∅)) → (𝑥 ∈ (On ∖ 1o) ∧ 𝐴 = (ω ·o 𝑥)))
11295, 111mpdan 687 . . . . . . . . . 10 (((𝐴 ∈ On ∧ Lim 𝐴) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ((ω ·o 𝑥) +o 𝑦) = 𝐴) → (𝑥 ∈ (On ∖ 1o) ∧ 𝐴 = (ω ·o 𝑥)))
1131123exp 1118 . . . . . . . . 9 ((𝐴 ∈ On ∧ Lim 𝐴) → ((𝑥 ∈ On ∧ 𝑦 ∈ ω) → (((ω ·o 𝑥) +o 𝑦) = 𝐴 → (𝑥 ∈ (On ∖ 1o) ∧ 𝐴 = (ω ·o 𝑥)))))
114113expdimp 452 . . . . . . . 8 (((𝐴 ∈ On ∧ Lim 𝐴) ∧ 𝑥 ∈ On) → (𝑦 ∈ ω → (((ω ·o 𝑥) +o 𝑦) = 𝐴 → (𝑥 ∈ (On ∖ 1o) ∧ 𝐴 = (ω ·o 𝑥)))))
115114rexlimdv 3151 . . . . . . 7 (((𝐴 ∈ On ∧ Lim 𝐴) ∧ 𝑥 ∈ On) → (∃𝑦 ∈ ω ((ω ·o 𝑥) +o 𝑦) = 𝐴 → (𝑥 ∈ (On ∖ 1o) ∧ 𝐴 = (ω ·o 𝑥))))
116115expimpd 453 . . . . . 6 ((𝐴 ∈ On ∧ Lim 𝐴) → ((𝑥 ∈ On ∧ ∃𝑦 ∈ ω ((ω ·o 𝑥) +o 𝑦) = 𝐴) → (𝑥 ∈ (On ∖ 1o) ∧ 𝐴 = (ω ·o 𝑥))))
117116reximdv2 3162 . . . . 5 ((𝐴 ∈ On ∧ Lim 𝐴) → (∃𝑥 ∈ On ∃𝑦 ∈ ω ((ω ·o 𝑥) +o 𝑦) = 𝐴 → ∃𝑥 ∈ (On ∖ 1o)𝐴 = (ω ·o 𝑥)))
11823, 117mpd 15 . . . 4 ((𝐴 ∈ On ∧ Lim 𝐴) → ∃𝑥 ∈ (On ∖ 1o)𝐴 = (ω ·o 𝑥))
119 simpr 484 . . . . . . 7 ((𝑥 ∈ (On ∖ 1o) ∧ 𝐴 = (ω ·o 𝑥)) → 𝐴 = (ω ·o 𝑥))
120 eldifi 4141 . . . . . . . . 9 (𝑥 ∈ (On ∖ 1o) → 𝑥 ∈ On)
12115, 120, 74sylancr 587 . . . . . . . 8 (𝑥 ∈ (On ∖ 1o) → (ω ·o 𝑥) ∈ On)
122121adantr 480 . . . . . . 7 ((𝑥 ∈ (On ∖ 1o) ∧ 𝐴 = (ω ·o 𝑥)) → (ω ·o 𝑥) ∈ On)
123119, 122eqeltrd 2839 . . . . . 6 ((𝑥 ∈ (On ∖ 1o) ∧ 𝐴 = (ω ·o 𝑥)) → 𝐴 ∈ On)
124 limom 7903 . . . . . . . . . . 11 Lim ω
12515, 124pm3.2i 470 . . . . . . . . . 10 (ω ∈ On ∧ Lim ω)
126 omlimcl2 43231 . . . . . . . . . 10 (((𝑥 ∈ On ∧ (ω ∈ On ∧ Lim ω)) ∧ ∅ ∈ 𝑥) → Lim (ω ·o 𝑥))
127125, 126mpanl2 701 . . . . . . . . 9 ((𝑥 ∈ On ∧ ∅ ∈ 𝑥) → Lim (ω ·o 𝑥))
128100, 127sylbi 217 . . . . . . . 8 (𝑥 ∈ (On ∖ 1o) → Lim (ω ·o 𝑥))
129128adantr 480 . . . . . . 7 ((𝑥 ∈ (On ∖ 1o) ∧ 𝐴 = (ω ·o 𝑥)) → Lim (ω ·o 𝑥))
130 limeq 6398 . . . . . . . 8 (𝐴 = (ω ·o 𝑥) → (Lim 𝐴 ↔ Lim (ω ·o 𝑥)))
131130adantl 481 . . . . . . 7 ((𝑥 ∈ (On ∖ 1o) ∧ 𝐴 = (ω ·o 𝑥)) → (Lim 𝐴 ↔ Lim (ω ·o 𝑥)))
132129, 131mpbird 257 . . . . . 6 ((𝑥 ∈ (On ∖ 1o) ∧ 𝐴 = (ω ·o 𝑥)) → Lim 𝐴)
133123, 132jca 511 . . . . 5 ((𝑥 ∈ (On ∖ 1o) ∧ 𝐴 = (ω ·o 𝑥)) → (𝐴 ∈ On ∧ Lim 𝐴))
134133rexlimiva 3145 . . . 4 (∃𝑥 ∈ (On ∖ 1o)𝐴 = (ω ·o 𝑥) → (𝐴 ∈ On ∧ Lim 𝐴))
135118, 134impbii 209 . . 3 ((𝐴 ∈ On ∧ Lim 𝐴) ↔ ∃𝑥 ∈ (On ∖ 1o)𝐴 = (ω ·o 𝑥))
136135orbi2i 912 . 2 ((𝐴 = On ∨ (𝐴 ∈ On ∧ Lim 𝐴)) ↔ (𝐴 = On ∨ ∃𝑥 ∈ (On ∖ 1o)𝐴 = (ω ·o 𝑥)))
1378, 13, 1363bitr2i 299 1 (Lim 𝐴 ↔ (𝐴 = On ∨ ∃𝑥 ∈ (On ∖ 1o)𝐴 = (ω ·o 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3o 1085  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068  Vcvv 3478  cdif 3960  c0 4339   cuni 4912  Ord word 6385  Oncon0 6386  Lim wlim 6387  suc csuc 6388  (class class class)co 7431  ωcom 7887  1oc1o 8498   +o coa 8502   ·o comu 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510
This theorem is referenced by: (None)
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