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 7654 | . 2 ⊢ Ord ω | |
2 | ordeleqon 7566 | . . 3 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On)) | |
3 | ordirr 6231 | . . . . . . 7 ⊢ (Ord ω → ¬ ω ∈ ω) | |
4 | 1, 3 | ax-mp 5 | . . . . . 6 ⊢ ¬ ω ∈ ω |
5 | elom 7647 | . . . . . . 7 ⊢ (ω ∈ ω ↔ (ω ∈ On ∧ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))) | |
6 | 5 | baib 539 | . . . . . 6 ⊢ (ω ∈ On → (ω ∈ ω ↔ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))) |
7 | 4, 6 | mtbii 329 | . . . . 5 ⊢ (ω ∈ On → ¬ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)) |
8 | limomss 7649 | . . . . . . . . . . 11 ⊢ (Lim 𝑥 → ω ⊆ 𝑥) | |
9 | limord 6272 | . . . . . . . . . . . 12 ⊢ (Lim 𝑥 → Ord 𝑥) | |
10 | ordsseleq 6242 | . . . . . . . . . . . 12 ⊢ ((Ord ω ∧ Ord 𝑥) → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥))) | |
11 | 1, 9, 10 | sylancr 590 | . . . . . . . . . . 11 ⊢ (Lim 𝑥 → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥))) |
12 | 8, 11 | mpbid 235 | . . . . . . . . . 10 ⊢ (Lim 𝑥 → (ω ∈ 𝑥 ∨ ω = 𝑥)) |
13 | 12 | ord 864 | . . . . . . . . 9 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → ω = 𝑥)) |
14 | limeq 6225 | . . . . . . . . . 10 ⊢ (ω = 𝑥 → (Lim ω ↔ Lim 𝑥)) | |
15 | 14 | biimprcd 253 | . . . . . . . . 9 ⊢ (Lim 𝑥 → (ω = 𝑥 → Lim ω)) |
16 | 13, 15 | syld 47 | . . . . . . . 8 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → Lim ω)) |
17 | 16 | con1d 147 | . . . . . . 7 ⊢ (Lim 𝑥 → (¬ Lim ω → ω ∈ 𝑥)) |
18 | 17 | com12 32 | . . . . . 6 ⊢ (¬ Lim ω → (Lim 𝑥 → ω ∈ 𝑥)) |
19 | 18 | alrimiv 1935 | . . . . 5 ⊢ (¬ Lim ω → ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)) |
20 | 7, 19 | nsyl2 143 | . . . 4 ⊢ (ω ∈ On → Lim ω) |
21 | limon 7615 | . . . . 5 ⊢ Lim On | |
22 | limeq 6225 | . . . . 5 ⊢ (ω = On → (Lim ω ↔ Lim On)) | |
23 | 21, 22 | mpbiri 261 | . . . 4 ⊢ (ω = On → Lim ω) |
24 | 20, 23 | jaoi 857 | . . 3 ⊢ ((ω ∈ On ∨ ω = On) → Lim ω) |
25 | 2, 24 | sylbi 220 | . 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 209 ∨ wo 847 ∀wal 1541 = wceq 1543 ∈ wcel 2110 ⊆ wss 3866 Ord word 6212 Oncon0 6213 Lim wlim 6214 ωcom 7644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-tr 5162 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-om 7645 |
This theorem is referenced by: peano2b 7661 ssnlim 7664 peano1 7667 onesuc 8257 oaabslem 8372 oaabs2 8374 omabslem 8375 infensuc 8824 infeq5i 9251 elom3 9263 omenps 9270 omensuc 9271 infdifsn 9272 cardlim 9588 r1om 9858 cfom 9878 ominf4 9926 alephom 10199 wunex3 10355 satom 33031 fmla 33056 exrecfnlem 35287 dfom6 40823 |
Copyright terms: Public domain | W3C validator |