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| Mirrors > Home > MPE Home > Th. List > trom | Structured version Visualization version GIF version | ||
| Description: The class of finite ordinals ω is a transitive class. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
| Ref | Expression |
|---|---|
| trom | ⊢ Tr ω |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dftr2 5261 | . 2 ⊢ (Tr ω ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑦 ∈ ω)) | |
| 2 | onelon 6409 | . . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ On) | |
| 3 | 2 | expcom 413 | . . . . . 6 ⊢ (𝑦 ∈ 𝑥 → (𝑥 ∈ On → 𝑦 ∈ On)) |
| 4 | limord 6444 | . . . . . . . . . . 11 ⊢ (Lim 𝑧 → Ord 𝑧) | |
| 5 | ordtr 6398 | . . . . . . . . . . 11 ⊢ (Ord 𝑧 → Tr 𝑧) | |
| 6 | trel 5268 | . . . . . . . . . . 11 ⊢ (Tr 𝑧 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑧)) | |
| 7 | 4, 5, 6 | 3syl 18 | . . . . . . . . . 10 ⊢ (Lim 𝑧 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑧)) |
| 8 | 7 | expd 415 | . . . . . . . . 9 ⊢ (Lim 𝑧 → (𝑦 ∈ 𝑥 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))) |
| 9 | 8 | com12 32 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝑥 → (Lim 𝑧 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))) |
| 10 | 9 | a2d 29 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑥 → ((Lim 𝑧 → 𝑥 ∈ 𝑧) → (Lim 𝑧 → 𝑦 ∈ 𝑧))) |
| 11 | 10 | alimdv 1916 | . . . . . 6 ⊢ (𝑦 ∈ 𝑥 → (∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧) → ∀𝑧(Lim 𝑧 → 𝑦 ∈ 𝑧))) |
| 12 | 3, 11 | anim12d 609 | . . . . 5 ⊢ (𝑦 ∈ 𝑥 → ((𝑥 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧)) → (𝑦 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑦 ∈ 𝑧)))) |
| 13 | elom 7890 | . . . . 5 ⊢ (𝑥 ∈ ω ↔ (𝑥 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑥 ∈ 𝑧))) | |
| 14 | elom 7890 | . . . . 5 ⊢ (𝑦 ∈ ω ↔ (𝑦 ∈ On ∧ ∀𝑧(Lim 𝑧 → 𝑦 ∈ 𝑧))) | |
| 15 | 12, 13, 14 | 3imtr4g 296 | . . . 4 ⊢ (𝑦 ∈ 𝑥 → (𝑥 ∈ ω → 𝑦 ∈ ω)) |
| 16 | 15 | imp 406 | . . 3 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑦 ∈ ω) |
| 17 | 16 | ax-gen 1795 | . 2 ⊢ ∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ ω) → 𝑦 ∈ ω) |
| 18 | 1, 17 | mpgbir 1799 | 1 ⊢ Tr ω |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 ∈ wcel 2108 Tr wtr 5259 Ord word 6383 Oncon0 6384 Lim wlim 6385 ωcom 7887 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-lim 6389 df-om 7888 |
| This theorem is referenced by: ordom 7897 elnn 7898 omsinds 7908 |
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