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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 6414 | . . . 4 ⊢ ¬ Lim ∅ | |
2 | limeq 6367 | . . . 4 ⊢ (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅)) | |
3 | 1, 2 | mtbiri 327 | . . 3 ⊢ (𝐴 = ∅ → ¬ Lim 𝐴) |
4 | 3 | necon2ai 2962 | . 2 ⊢ (Lim 𝐴 → 𝐴 ≠ ∅) |
5 | nlim1 8485 | . . . 4 ⊢ ¬ Lim 1o | |
6 | limeq 6367 | . . . 4 ⊢ (𝐴 = 1o → (Lim 𝐴 ↔ Lim 1o)) | |
7 | 5, 6 | mtbiri 327 | . . 3 ⊢ (𝐴 = 1o → ¬ Lim 𝐴) |
8 | 7 | necon2ai 2962 | . 2 ⊢ (Lim 𝐴 → 𝐴 ≠ 1o) |
9 | nlim2 8486 | . . . 4 ⊢ ¬ Lim 2o | |
10 | limeq 6367 | . . . 4 ⊢ (𝐴 = 2o → (Lim 𝐴 ↔ Lim 2o)) | |
11 | 9, 10 | mtbiri 327 | . . 3 ⊢ (𝐴 = 2o → ¬ Lim 𝐴) |
12 | 11 | necon2ai 2962 | . 2 ⊢ (Lim 𝐴 → 𝐴 ≠ 2o) |
13 | limord 6415 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
14 | ord2eln012 8493 | . . 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 1339 | 1 ⊢ (Lim 𝐴 → 2o ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∅c0 4315 Ord word 6354 Lim wlim 6356 1oc1o 8455 2oc2o 8456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-tr 5257 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-1o 8462 df-2o 8463 |
This theorem is referenced by: 2onn 8638 |
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