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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 6382 | . 2 ⊢ (Lim 𝐴 → Ord 𝐴) | |
2 | ordeleqon 7721 | . . 3 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
3 | elom 7810 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))) | |
4 | 3 | simprbi 498 | . . . . . . . . 9 ⊢ (𝑥 ∈ ω → ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)) |
5 | limeq 6334 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (Lim 𝑦 ↔ Lim 𝐴)) | |
6 | eleq2 2827 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴)) | |
7 | 5, 6 | imbi12d 345 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((Lim 𝑦 → 𝑥 ∈ 𝑦) ↔ (Lim 𝐴 → 𝑥 ∈ 𝐴))) |
8 | 7 | spcgv 3558 | . . . . . . . . 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 408 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑥 ∈ ω → 𝑥 ∈ 𝐴)) |
12 | 11 | ssrdv 3955 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ω ⊆ 𝐴) |
13 | 12 | ex 414 | . . . 4 ⊢ (𝐴 ∈ On → (Lim 𝐴 → ω ⊆ 𝐴)) |
14 | omsson 7811 | . . . . . 6 ⊢ ω ⊆ On | |
15 | sseq2 3975 | . . . . . 6 ⊢ (𝐴 = On → (ω ⊆ 𝐴 ↔ ω ⊆ On)) | |
16 | 14, 15 | mpbiri 258 | . . . . 5 ⊢ (𝐴 = On → ω ⊆ 𝐴) |
17 | 16 | a1d 25 | . . . 4 ⊢ (𝐴 = On → (Lim 𝐴 → ω ⊆ 𝐴)) |
18 | 13, 17 | jaoi 856 | . . 3 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (Lim 𝐴 → ω ⊆ 𝐴)) |
19 | 2, 18 | sylbi 216 | . 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 ∧ wa 397 ∨ wo 846 ∀wal 1540 = wceq 1542 ∈ wcel 2107 ⊆ wss 3915 Ord word 6321 Oncon0 6322 Lim wlim 6323 ωcom 7807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-tr 5228 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-ord 6325 df-on 6326 df-lim 6327 df-om 7808 |
This theorem is referenced by: limom 7823 rdg0 8372 frfnom 8386 frsuc 8388 r1fin 9716 rankdmr1 9744 rankeq0b 9803 cardlim 9915 ackbij2 10186 cfom 10207 wunom 10663 inar1 10718 bj-rdg0gALT 35571 |
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