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Theorem dflim5 43347
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 6443 . . . . 5 (Lim 𝐴 → Ord 𝐴)
2 ordeleqon 7803 . . . . . . 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 7857 . . . . 5 Lim On
10 limeq 6395 . . . . 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 9687 . . . . . . . 8 ω ∈ On
1615a1i 11 . . . . . . 7 (𝐴 ∈ On → ω ∈ On)
17 id 22 . . . . . . 7 (𝐴 ∈ On → 𝐴 ∈ On)
18 peano1 7911 . . . . . . . . 9 ∅ ∈ ω
1918ne0ii 4343 . . . . . . . 8 ω ≠ ∅
2019a1i 11 . . . . . . 7 (𝐴 ∈ On → ω ≠ ∅)
2116, 17, 203jca 1128 . . . . . 6 (𝐴 ∈ On → (ω ∈ On ∧ 𝐴 ∈ On ∧ ω ≠ ∅))
22 omeulem1 8621 . . . . . 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 6395 . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → 𝑦 ∈ ω)
27 nnlim 7902 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ ω → ¬ Lim 𝑦)
2826, 27syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ ∅ ∈ 𝑥) → ¬ Lim 𝑦)
29 on0eln0 6439 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ On → (∅ ∈ 𝑥𝑥 ≠ ∅))
3029biimprd 248 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 ∈ On → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
3130necon1bd 2957 . . . . . . . . . . . . . . . . . . . . . . . 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 7448 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (ω ·o 𝑥) = (ω ·o ∅))
37 om0 8556 . . . . . . . . . . . . . . . . . . . . . . . . 25 (ω ∈ On → (ω ·o ∅) = ∅)
3815, 37mp1i 13 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (ω ·o ∅) = ∅)
3936, 38eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (ω ·o 𝑥) = ∅)
4039oveq1d 7447 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → ((ω ·o 𝑥) +o 𝑦) = (∅ +o 𝑦))
41 nna0r 8648 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ ω → (∅ +o 𝑦) = 𝑦)
4241adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → (∅ +o 𝑦) = 𝑦)
4340, 42eqtrd 2776 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 = ∅ ∧ 𝑦 ∈ ω) → ((ω ·o 𝑥) +o 𝑦) = 𝑦)
44 limeq 6395 . . . . . . . . . . . . . . . . . . . . 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 7465 . . . . . . . . . . . . . . . . . . . . 21 ((ω ·o 𝑥) +o 𝑦) ∈ V
49 nlimsucg 7864 . . . . . . . . . . . . . . . . . . . . 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 7896 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ω → Ord 𝑦)
52 orduniorsuc 7851 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6388 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2943 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1430 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7448 . . . . . . . . . . . . . . . . . . . . . 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 8575 . . . . . . . . . . . . . . . . . . . . . . . 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 7894 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ ω → 𝑦 ∈ On)
77 onuni 7809 . . . . . . . . . . . . . . . . . . . . . . . . . 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 8563 . . . . . . . . . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥 ∈ On ∧ 𝑦 ∈ ω) ∧ ¬ 𝑦 = ∅) → ((ω ·o 𝑥) +o 𝑦) = suc ((ω ·o 𝑥) +o 𝑦))
84 limeq 6395 . . . . . . . . . . . . . . . . . . . . 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 1137 . . . . . . . . . . . . . . 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 8540 . . . . . . . . . . . . 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 7448 . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . 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 8555 . . . . . . . . . . . . . . 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 2784 . . . . . . . . . . . 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 1119 . . . . . . . . 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 3152 . . . . . . 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 3163 . . . . 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 4130 . . . . . . . . 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 2840 . . . . . 6 ((𝑥 ∈ (On ∖ 1o) ∧ 𝐴 = (ω ·o 𝑥)) → 𝐴 ∈ On)
124 limom 7904 . . . . . . . . . . 11 Lim ω
12515, 124pm3.2i 470 . . . . . . . . . 10 (ω ∈ On ∧ Lim ω)
126 omlimcl2 43259 . . . . . . . . . 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 6395 . . . . . . . 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 3146 . . . 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 1539  wcel 2107  wne 2939  wrex 3069  Vcvv 3479  cdif 3947  c0 4332   cuni 4906  Ord word 6382  Oncon0 6383  Lim wlim 6384  suc csuc 6385  (class class class)co 7432  ωcom 7888  1oc1o 8500   +o coa 8504   ·o comu 8505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-omul 8512
This theorem is referenced by: (None)
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