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| 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 6443 | . . . 4 ⊢ ¬ Lim ∅ | |
| 2 | limeq 6396 | . . . 4 ⊢ (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅)) | |
| 3 | 1, 2 | mtbiri 327 | . . 3 ⊢ (𝐴 = ∅ → ¬ Lim 𝐴) | 
| 4 | 3 | necon2ai 2970 | . 2 ⊢ (Lim 𝐴 → 𝐴 ≠ ∅) | 
| 5 | nlim1 8527 | . . . 4 ⊢ ¬ Lim 1o | |
| 6 | limeq 6396 | . . . 4 ⊢ (𝐴 = 1o → (Lim 𝐴 ↔ Lim 1o)) | |
| 7 | 5, 6 | mtbiri 327 | . . 3 ⊢ (𝐴 = 1o → ¬ Lim 𝐴) | 
| 8 | 7 | necon2ai 2970 | . 2 ⊢ (Lim 𝐴 → 𝐴 ≠ 1o) | 
| 9 | nlim2 8528 | . . . 4 ⊢ ¬ Lim 2o | |
| 10 | limeq 6396 | . . . 4 ⊢ (𝐴 = 2o → (Lim 𝐴 ↔ Lim 2o)) | |
| 11 | 9, 10 | mtbiri 327 | . . 3 ⊢ (𝐴 = 2o → ¬ Lim 𝐴) | 
| 12 | 11 | necon2ai 2970 | . 2 ⊢ (Lim 𝐴 → 𝐴 ≠ 2o) | 
| 13 | limord 6444 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
| 14 | ord2eln012 8535 | . . 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 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 Ord word 6383 Lim wlim 6385 1oc1o 8499 2oc2o 8500 | 
| 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 | 
| 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-1o 8506 df-2o 8507 | 
| This theorem is referenced by: 2onn 8680 | 
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