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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 6376 | . 2 ⊢ (Lim 𝐴 → Ord 𝐴) | |
2 | ordeleqon 7713 | . . 3 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
3 | elom 7802 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))) | |
4 | 3 | simprbi 497 | . . . . . . . . 9 ⊢ (𝑥 ∈ ω → ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)) |
5 | limeq 6328 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (Lim 𝑦 ↔ Lim 𝐴)) | |
6 | eleq2 2826 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴)) | |
7 | 5, 6 | imbi12d 344 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((Lim 𝑦 → 𝑥 ∈ 𝑦) ↔ (Lim 𝐴 → 𝑥 ∈ 𝐴))) |
8 | 7 | spcgv 3554 | . . . . . . . . 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 407 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑥 ∈ ω → 𝑥 ∈ 𝐴)) |
12 | 11 | ssrdv 3949 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ω ⊆ 𝐴) |
13 | 12 | ex 413 | . . . 4 ⊢ (𝐴 ∈ On → (Lim 𝐴 → ω ⊆ 𝐴)) |
14 | omsson 7803 | . . . . . 6 ⊢ ω ⊆ On | |
15 | sseq2 3969 | . . . . . 6 ⊢ (𝐴 = On → (ω ⊆ 𝐴 ↔ ω ⊆ On)) | |
16 | 14, 15 | mpbiri 257 | . . . . 5 ⊢ (𝐴 = On → ω ⊆ 𝐴) |
17 | 16 | a1d 25 | . . . 4 ⊢ (𝐴 = On → (Lim 𝐴 → ω ⊆ 𝐴)) |
18 | 13, 17 | jaoi 855 | . . 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 396 ∨ wo 845 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ⊆ wss 3909 Ord word 6315 Oncon0 6316 Lim wlim 6317 ωcom 7799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-tr 5222 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-ord 6319 df-on 6320 df-lim 6321 df-om 7800 |
This theorem is referenced by: limom 7815 rdg0 8364 frfnom 8378 frsuc 8380 r1fin 9706 rankdmr1 9734 rankeq0b 9793 cardlim 9905 ackbij2 10176 cfom 10197 wunom 10653 inar1 10708 bj-rdg0gALT 35531 |
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