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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 4624 | . 2 ⊢ Ord ω | |
2 | peano1 4611 | . 2 ⊢ ∅ ∈ ω | |
3 | vex 2755 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
4 | 3 | sucex 4516 | . . . . . . . 8 ⊢ suc 𝑥 ∈ V |
5 | 4 | isseti 2760 | . . . . . . 7 ⊢ ∃𝑧 𝑧 = suc 𝑥 |
6 | peano2 4612 | . . . . . . . . 9 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω) | |
7 | 3 | sucid 4435 | . . . . . . . . 9 ⊢ 𝑥 ∈ suc 𝑥 |
8 | 6, 7 | jctil 312 | . . . . . . . 8 ⊢ (𝑥 ∈ ω → (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω)) |
9 | eleq2 2253 | . . . . . . . . 9 ⊢ (𝑧 = suc 𝑥 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ suc 𝑥)) | |
10 | eleq1 2252 | . . . . . . . . 9 ⊢ (𝑧 = suc 𝑥 → (𝑧 ∈ ω ↔ suc 𝑥 ∈ ω)) | |
11 | 9, 10 | anbi12d 473 | . . . . . . . 8 ⊢ (𝑧 = suc 𝑥 → ((𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω) ↔ (𝑥 ∈ suc 𝑥 ∧ suc 𝑥 ∈ ω))) |
12 | 8, 11 | imbitrrid 156 | . . . . . . 7 ⊢ (𝑧 = suc 𝑥 → (𝑥 ∈ ω → (𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω))) |
13 | 5, 12 | eximii 1613 | . . . . . 6 ⊢ ∃𝑧(𝑥 ∈ ω → (𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω)) |
14 | 13 | 19.37aiv 1686 | . . . . 5 ⊢ (𝑥 ∈ ω → ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω)) |
15 | eluni 3827 | . . . . 5 ⊢ (𝑥 ∈ ∪ ω ↔ ∃𝑧(𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ω)) | |
16 | 14, 15 | sylibr 134 | . . . 4 ⊢ (𝑥 ∈ ω → 𝑥 ∈ ∪ ω) |
17 | 16 | ssriv 3174 | . . 3 ⊢ ω ⊆ ∪ ω |
18 | orduniss 4443 | . . . 4 ⊢ (Ord ω → ∪ ω ⊆ ω) | |
19 | 1, 18 | ax-mp 5 | . . 3 ⊢ ∪ ω ⊆ ω |
20 | 17, 19 | eqssi 3186 | . 2 ⊢ ω = ∪ ω |
21 | dflim2 4388 | . 2 ⊢ (Lim ω ↔ (Ord ω ∧ ∅ ∈ ω ∧ ω = ∪ ω)) | |
22 | 1, 2, 20, 21 | mpbir3an 1181 | 1 ⊢ Lim ω |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∃wex 1503 ∈ wcel 2160 ⊆ wss 3144 ∅c0 3437 ∪ cuni 3824 Ord word 4380 Lim wlim 4382 suc csuc 4383 ωcom 4607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-int 3860 df-tr 4117 df-iord 4384 df-ilim 4387 df-suc 4389 df-iom 4608 |
This theorem is referenced by: freccllem 6428 frecfcllem 6430 frecsuclem 6432 |
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