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| 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. Theorem 1.23 of [Schloeder] p. 4. 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 7897 | . 2 ⊢ Ord ω | |
| 2 | ordeleqon 7802 | . . 3 ⊢ (Ord ω ↔ (ω ∈ On ∨ ω = On)) | |
| 3 | ordirr 6402 | . . . . . . 7 ⊢ (Ord ω → ¬ ω ∈ ω) | |
| 4 | 1, 3 | ax-mp 5 | . . . . . 6 ⊢ ¬ ω ∈ ω |
| 5 | elom 7890 | . . . . . . 7 ⊢ (ω ∈ ω ↔ (ω ∈ On ∧ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))) | |
| 6 | 5 | baib 535 | . . . . . 6 ⊢ (ω ∈ On → (ω ∈ ω ↔ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥))) |
| 7 | 4, 6 | mtbii 326 | . . . . 5 ⊢ (ω ∈ On → ¬ ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)) |
| 8 | limomss 7892 | . . . . . . . . . . 11 ⊢ (Lim 𝑥 → ω ⊆ 𝑥) | |
| 9 | limord 6444 | . . . . . . . . . . . 12 ⊢ (Lim 𝑥 → Ord 𝑥) | |
| 10 | ordsseleq 6413 | . . . . . . . . . . . 12 ⊢ ((Ord ω ∧ Ord 𝑥) → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥))) | |
| 11 | 1, 9, 10 | sylancr 587 | . . . . . . . . . . 11 ⊢ (Lim 𝑥 → (ω ⊆ 𝑥 ↔ (ω ∈ 𝑥 ∨ ω = 𝑥))) |
| 12 | 8, 11 | mpbid 232 | . . . . . . . . . 10 ⊢ (Lim 𝑥 → (ω ∈ 𝑥 ∨ ω = 𝑥)) |
| 13 | 12 | ord 865 | . . . . . . . . 9 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → ω = 𝑥)) |
| 14 | limeq 6396 | . . . . . . . . . 10 ⊢ (ω = 𝑥 → (Lim ω ↔ Lim 𝑥)) | |
| 15 | 14 | biimprcd 250 | . . . . . . . . 9 ⊢ (Lim 𝑥 → (ω = 𝑥 → Lim ω)) |
| 16 | 13, 15 | syld 47 | . . . . . . . 8 ⊢ (Lim 𝑥 → (¬ ω ∈ 𝑥 → Lim ω)) |
| 17 | 16 | con1d 145 | . . . . . . 7 ⊢ (Lim 𝑥 → (¬ Lim ω → ω ∈ 𝑥)) |
| 18 | 17 | com12 32 | . . . . . 6 ⊢ (¬ Lim ω → (Lim 𝑥 → ω ∈ 𝑥)) |
| 19 | 18 | alrimiv 1927 | . . . . 5 ⊢ (¬ Lim ω → ∀𝑥(Lim 𝑥 → ω ∈ 𝑥)) |
| 20 | 7, 19 | nsyl2 141 | . . . 4 ⊢ (ω ∈ On → Lim ω) |
| 21 | limon 7856 | . . . . 5 ⊢ Lim On | |
| 22 | limeq 6396 | . . . . 5 ⊢ (ω = On → (Lim ω ↔ Lim On)) | |
| 23 | 21, 22 | mpbiri 258 | . . . 4 ⊢ (ω = On → Lim ω) |
| 24 | 20, 23 | jaoi 858 | . . 3 ⊢ ((ω ∈ On ∨ ω = On) → Lim ω) |
| 25 | 2, 24 | sylbi 217 | . 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 206 ∨ wo 848 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 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 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 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-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 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-suc 6390 df-om 7888 |
| This theorem is referenced by: peano2b 7904 ssnlim 7907 peano1OLD 7911 onesuc 8568 oaabslem 8685 oaabs2 8687 omabslem 8688 infensuc 9195 infeq5i 9676 elom3 9688 omenps 9695 omensuc 9696 infdifsn 9697 cardlim 10012 r1om 10283 cfom 10304 ominf4 10352 alephom 10625 wunex3 10781 satom 35361 fmla 35386 exrecfnlem 37380 onexlimgt 43255 oaabsb 43307 nnoeomeqom 43325 succlg 43341 dflim5 43342 dfom6 43544 |
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