Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > limom | Structured version Visualization version GIF version |
Description: Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. Our proof, however, does not require the Axiom of Infinity. (Contributed by NM, 26-Mar-1995.) (Proof shortened by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
limom | ⊢ Lim ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordom 6966 | . 2 ⊢ Ord ω | |
2 | ordeleqon 6880 | . . 3 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On)) | |
3 | ordirr 5658 | . . . . . . 7 ⊢ (Ord ω → ¬ ω ∈ ω) | |
4 | 1, 3 | ax-mp 5 | . . . . . 6 ⊢ ¬ ω ∈ ω |
5 | elom 6960 | . . . . . . 7 ⊢ (ω ∈ ω ↔ (ω ∈ On ∧ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))) | |
6 | 5 | baib 942 | . . . . . 6 ⊢ (ω ∈ On → (ω ∈ ω ↔ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))) |
7 | 4, 6 | mtbii 315 | . . . . 5 ⊢ (ω ∈ On → ¬ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)) |
8 | limomss 6962 | . . . . . . . . . . 11 ⊢ (Lim 𝑥 → ω ⊆ 𝑥) | |
9 | limord 5701 | . . . . . . . . . . . 12 ⊢ (Lim 𝑥 → Ord 𝑥) | |
10 | ordsseleq 5669 | . . . . . . . . . . . 12 ⊢ ((Ord ω ∧ Ord 𝑥) → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥))) | |
11 | 1, 9, 10 | sylancr 694 | . . . . . . . . . . 11 ⊢ (Lim 𝑥 → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥))) |
12 | 8, 11 | mpbid 221 | . . . . . . . . . 10 ⊢ (Lim 𝑥 → (ω ∈ 𝑥 ∨ ω = 𝑥)) |
13 | 12 | ord 391 | . . . . . . . . 9 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → ω = 𝑥)) |
14 | limeq 5652 | . . . . . . . . . 10 ⊢ (ω = 𝑥 → (Lim ω ↔ Lim 𝑥)) | |
15 | 14 | biimprcd 239 | . . . . . . . . 9 ⊢ (Lim 𝑥 → (ω = 𝑥 → Lim ω)) |
16 | 13, 15 | syld 46 | . . . . . . . 8 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → Lim ω)) |
17 | 16 | con1d 138 | . . . . . . 7 ⊢ (Lim 𝑥 → (¬ Lim ω → ω ∈ 𝑥)) |
18 | 17 | com12 32 | . . . . . 6 ⊢ (¬ Lim ω → (Lim 𝑥 → ω ∈ 𝑥)) |
19 | 18 | alrimiv 1842 | . . . . 5 ⊢ (¬ Lim ω → ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)) |
20 | 7, 19 | nsyl2 141 | . . . 4 ⊢ (ω ∈ On → Lim ω) |
21 | limon 6928 | . . . . 5 ⊢ Lim On | |
22 | limeq 5652 | . . . . 5 ⊢ (ω = On → (Lim ω ↔ Lim On)) | |
23 | 21, 22 | mpbiri 247 | . . . 4 ⊢ (ω = On → Lim ω) |
24 | 20, 23 | jaoi 393 | . . 3 ⊢ ((ω ∈ On ∨ ω = On) → Lim ω) |
25 | 2, 24 | sylbi 206 | . 2 ⊢ (Ord ω → Lim ω) |
26 | 1, 25 | ax-mp 5 | 1 ⊢ Lim ω |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∀wal 1473 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 Ord word 5639 Oncon0 5640 Lim wlim 5641 ωcom 6957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-om 6958 |
This theorem is referenced by: peano2b 6973 ssnlim 6975 peano1 6977 onesuc 7497 oaabslem 7610 oaabs2 7612 omabslem 7613 infensuc 8023 infeq5i 8416 elom3 8428 omenps 8435 omensuc 8436 infdifsn 8437 cardlim 8681 r1om 8949 cfom 8969 ominf4 9017 alephom 9286 wunex3 9442 |
Copyright terms: Public domain | W3C validator |