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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 6378 | . 2 ⊢ (Lim 𝐴 → Ord 𝐴) | |
| 2 | ordeleqon 7727 | . . 3 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
| 3 | elom 7811 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦))) | |
| 4 | 3 | simprbi 496 | . . . . . . . . 9 ⊢ (𝑥 ∈ ω → ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)) |
| 5 | limeq 6329 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (Lim 𝑦 ↔ Lim 𝐴)) | |
| 6 | eleq2 2825 | . . . . . . . . . . 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 3550 | . . . . . . . . 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 406 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑥 ∈ ω → 𝑥 ∈ 𝐴)) |
| 12 | 11 | ssrdv 3939 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → ω ⊆ 𝐴) |
| 13 | 12 | ex 412 | . . . 4 ⊢ (𝐴 ∈ On → (Lim 𝐴 → ω ⊆ 𝐴)) |
| 14 | omsson 7812 | . . . . . 6 ⊢ ω ⊆ On | |
| 15 | sseq2 3960 | . . . . . 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 857 | . . 3 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (Lim 𝐴 → ω ⊆ 𝐴)) |
| 19 | 2, 18 | sylbi 217 | . 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 395 ∨ wo 847 ∀wal 1539 = wceq 1541 ∈ wcel 2113 ⊆ wss 3901 Ord word 6316 Oncon0 6317 Lim wlim 6318 ωcom 7808 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-sb 2068 df-clab 2715 df-cleq 2728 df-clel 2811 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-tr 5206 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-ord 6320 df-on 6321 df-lim 6322 df-om 7809 |
| This theorem is referenced by: limom 7824 rdg0 8352 frfnom 8366 frsuc 8368 r1fin 9685 rankdmr1 9713 rankeq0b 9772 cardlim 9884 ackbij2 10152 cfom 10174 wunom 10631 inar1 10686 bj-rdg0gALT 37272 |
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