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Mirrors > Home > MPE Home > Th. List > 2ellim | Structured version Visualization version GIF version |
Description: A limit ordinal contains 2. (Contributed by BTernaryTau, 1-Dec-2024.) |
Ref | Expression |
---|---|
2ellim | ⊢ (Lim 𝐴 → 2o ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nlim0 6415 | . . . 4 ⊢ ¬ Lim ∅ | |
2 | limeq 6368 | . . . 4 ⊢ (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅)) | |
3 | 1, 2 | mtbiri 327 | . . 3 ⊢ (𝐴 = ∅ → ¬ Lim 𝐴) |
4 | 3 | necon2ai 2971 | . 2 ⊢ (Lim 𝐴 → 𝐴 ≠ ∅) |
5 | nlim1 8476 | . . . 4 ⊢ ¬ Lim 1o | |
6 | limeq 6368 | . . . 4 ⊢ (𝐴 = 1o → (Lim 𝐴 ↔ Lim 1o)) | |
7 | 5, 6 | mtbiri 327 | . . 3 ⊢ (𝐴 = 1o → ¬ Lim 𝐴) |
8 | 7 | necon2ai 2971 | . 2 ⊢ (Lim 𝐴 → 𝐴 ≠ 1o) |
9 | nlim2 8477 | . . . 4 ⊢ ¬ Lim 2o | |
10 | limeq 6368 | . . . 4 ⊢ (𝐴 = 2o → (Lim 𝐴 ↔ Lim 2o)) | |
11 | 9, 10 | mtbiri 327 | . . 3 ⊢ (𝐴 = 2o → ¬ Lim 𝐴) |
12 | 11 | necon2ai 2971 | . 2 ⊢ (Lim 𝐴 → 𝐴 ≠ 2o) |
13 | limord 6416 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
14 | ord2eln012 8484 | . . 3 ⊢ (Ord 𝐴 → (2o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o ∧ 𝐴 ≠ 2o))) | |
15 | 13, 14 | syl 17 | . 2 ⊢ (Lim 𝐴 → (2o ∈ 𝐴 ↔ (𝐴 ≠ ∅ ∧ 𝐴 ≠ 1o ∧ 𝐴 ≠ 2o))) |
16 | 4, 8, 12, 15 | mpbir3and 1343 | 1 ⊢ (Lim 𝐴 → 2o ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∅c0 4320 Ord word 6355 Lim wlim 6357 1oc1o 8446 2oc2o 8447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-tr 5262 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-1o 8453 df-2o 8454 |
This theorem is referenced by: 2onn 8629 |
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