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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 6125 | . 2 ⊢ (Lim 𝐴 → Ord 𝐴) | |
2 | ordeleqon 7359 | . . 3 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
3 | elom 7439 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))) | |
4 | 3 | simprbi 497 | . . . . . . . . 9 ⊢ (𝑥 ∈ ω → ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)) |
5 | limeq 6078 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (Lim 𝑦 ↔ Lim 𝐴)) | |
6 | eleq2 2871 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴)) | |
7 | 5, 6 | imbi12d 346 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → ((Lim 𝑦 → 𝑥 ∈ 𝑦) ↔ (Lim 𝐴 → 𝑥 ∈ 𝐴))) |
8 | 7 | spcgv 3538 | . . . . . . . . 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 3895 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ω ⊆ 𝐴) |
13 | 12 | ex 413 | . . . 4 ⊢ (𝐴 ∈ On → (Lim 𝐴 → ω ⊆ 𝐴)) |
14 | omsson 7440 | . . . . . 6 ⊢ ω ⊆ On | |
15 | sseq2 3914 | . . . . . 6 ⊢ (𝐴 = On → (ω ⊆ 𝐴 ↔ ω ⊆ On)) | |
16 | 14, 15 | mpbiri 259 | . . . . 5 ⊢ (𝐴 = On → ω ⊆ 𝐴) |
17 | 16 | a1d 25 | . . . 4 ⊢ (𝐴 = On → (Lim 𝐴 → ω ⊆ 𝐴)) |
18 | 13, 17 | jaoi 852 | . . 3 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (Lim 𝐴 → ω ⊆ 𝐴)) |
19 | 2, 18 | sylbi 218 | . 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 842 ∀wal 1520 = wceq 1522 ∈ wcel 2081 ⊆ wss 3859 Ord word 6065 Oncon0 6066 Lim wlim 6067 ωcom 7436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-tr 5064 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-om 7437 |
This theorem is referenced by: limom 7451 rdg0 7909 frfnom 7922 frsuc 7924 r1fin 9048 rankdmr1 9076 rankeq0b 9135 cardlim 9247 ackbij2 9511 cfom 9532 wunom 9988 inar1 10043 |
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