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Mirrors > Home > ILE Home > Th. List > limom | GIF version |
Description: Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. (Contributed by NM, 26-Mar-1995.) (Proof rewritten by Jim Kingdon, 5-Jan-2019.) |
Ref | Expression |
---|---|
limom | ⊢ Lim ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordom 4458 | . 2 ⊢ Ord ω | |
2 | peano1 4446 | . 2 ⊢ ∅ ∈ ω | |
3 | vex 2644 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
4 | 3 | sucex 4353 | . . . . . . . 8 ⊢ suc 𝑥 ∈ V |
5 | 4 | isseti 2649 | . . . . . . 7 ⊢ ∃𝑧 𝑧 = suc 𝑥 |
6 | peano2 4447 | . . . . . . . . 9 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω) | |
7 | 3 | sucid 4277 | . . . . . . . . 9 ⊢ 𝑥 ∈ suc 𝑥 |
8 | 6, 7 | jctil 308 | . . . . . . . 8 ⊢ (𝑥 ∈ ω → (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω)) |
9 | eleq2 2163 | . . . . . . . . 9 ⊢ (𝑧 = suc 𝑥 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ suc 𝑥)) | |
10 | eleq1 2162 | . . . . . . . . 9 ⊢ (𝑧 = suc 𝑥 → (𝑧 ∈ ω ↔ suc 𝑥 ∈ ω)) | |
11 | 9, 10 | anbi12d 460 | . . . . . . . 8 ⊢ (𝑧 = suc 𝑥 → ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω) ↔ (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω))) |
12 | 8, 11 | syl5ibr 155 | . . . . . . 7 ⊢ (𝑧 = suc 𝑥 → (𝑥 ∈ ω → (𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω))) |
13 | 5, 12 | eximii 1549 | . . . . . 6 ⊢ ∃𝑧(𝑥 ∈ ω → (𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω)) |
14 | 13 | 19.37aiv 1621 | . . . . 5 ⊢ (𝑥 ∈ ω → ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω)) |
15 | eluni 3686 | . . . . 5 ⊢ (𝑥 ∈ ∪ ω ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω)) | |
16 | 14, 15 | sylibr 133 | . . . 4 ⊢ (𝑥 ∈ ω → 𝑥 ∈ ∪ ω) |
17 | 16 | ssriv 3051 | . . 3 ⊢ ω ⊆ ∪ ω |
18 | orduniss 4285 | . . . 4 ⊢ (Ord ω → ∪ ω ⊆ ω) | |
19 | 1, 18 | ax-mp 7 | . . 3 ⊢ ∪ ω ⊆ ω |
20 | 17, 19 | eqssi 3063 | . 2 ⊢ ω = ∪ ω |
21 | dflim2 4230 | . 2 ⊢ (Lim ω ↔ (Ord ω ∧ ∅ ∈ ω ∧ ω = ∪ ω)) | |
22 | 1, 2, 20, 21 | mpbir3an 1131 | 1 ⊢ Lim ω |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1299 ∃wex 1436 ∈ wcel 1448 ⊆ wss 3021 ∅c0 3310 ∪ cuni 3683 Ord word 4222 Lim wlim 4224 suc csuc 4225 ωcom 4442 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-iinf 4440 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-uni 3684 df-int 3719 df-tr 3967 df-iord 4226 df-ilim 4229 df-suc 4231 df-iom 4443 |
This theorem is referenced by: freccllem 6229 frecfcllem 6231 frecsuclem 6233 |
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