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Mirrors > Home > MPE Home > Th. List > limomss | Structured version Visualization version GIF version |
Description: The class of natural numbers is a subclass of any (infinite) limit ordinal. Exercise 1 of [TakeutiZaring] p. 44. Remarkably, our proof does not require the Axiom of Infinity. (Contributed by NM, 30-Oct-2003.) |
Ref | Expression |
---|---|
limomss | ⊢ (Lim 𝐴 → ω ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limord 5822 | . 2 ⊢ (Lim 𝐴 → Ord 𝐴) | |
2 | ordeleqon 7030 | . . 3 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
3 | elom 7110 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))) | |
4 | 3 | simprbi 479 | . . . . . . . . 9 ⊢ (𝑥 ∈ ω → ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)) |
5 | limeq 5773 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (Lim 𝑦 ↔ Lim 𝐴)) | |
6 | eleq2 2719 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴)) | |
7 | 5, 6 | imbi12d 333 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((Lim 𝑦 → 𝑥 ∈ 𝑦) ↔ (Lim 𝐴 → 𝑥 ∈ 𝐴))) |
8 | 7 | spcgv 3324 | . . . . . . . . 9 ⊢ (𝐴 ∈ On → (∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦) → (Lim 𝐴 → 𝑥 ∈ 𝐴))) |
9 | 4, 8 | syl5 34 | . . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑥 ∈ ω → (Lim 𝐴 → 𝑥 ∈ 𝐴))) |
10 | 9 | com23 86 | . . . . . . 7 ⊢ (𝐴 ∈ On → (Lim 𝐴 → (𝑥 ∈ ω → 𝑥 ∈ 𝐴))) |
11 | 10 | imp 444 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑥 ∈ ω → 𝑥 ∈ 𝐴)) |
12 | 11 | ssrdv 3642 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ω ⊆ 𝐴) |
13 | 12 | ex 449 | . . . 4 ⊢ (𝐴 ∈ On → (Lim 𝐴 → ω ⊆ 𝐴)) |
14 | omsson 7111 | . . . . . 6 ⊢ ω ⊆ On | |
15 | sseq2 3660 | . . . . . 6 ⊢ (𝐴 = On → (ω ⊆ 𝐴 ↔ ω ⊆ On)) | |
16 | 14, 15 | mpbiri 248 | . . . . 5 ⊢ (𝐴 = On → ω ⊆ 𝐴) |
17 | 16 | a1d 25 | . . . 4 ⊢ (𝐴 = On → (Lim 𝐴 → ω ⊆ 𝐴)) |
18 | 13, 17 | jaoi 393 | . . 3 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (Lim 𝐴 → ω ⊆ 𝐴)) |
19 | 2, 18 | sylbi 207 | . 2 ⊢ (Ord 𝐴 → (Lim 𝐴 → ω ⊆ 𝐴)) |
20 | 1, 19 | mpcom 38 | 1 ⊢ (Lim 𝐴 → ω ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 ∀wal 1521 = wceq 1523 ∈ wcel 2030 ⊆ wss 3607 Ord word 5760 Oncon0 5761 Lim wlim 5762 ωcom 7107 |
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-sep 4814 ax-nul 4822 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-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-tr 4786 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-om 7108 |
This theorem is referenced by: limom 7122 rdg0 7562 frfnom 7575 frsuc 7577 r1fin 8674 rankdmr1 8702 rankeq0b 8761 cardlim 8836 ackbij2 9103 cfom 9124 wunom 9580 inar1 9635 |
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