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Mirrors > Home > MPE Home > Th. List > Mathboxes > nlim2NEW | Structured version Visualization version GIF version |
Description: 2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.) |
Ref | Expression |
---|---|
nlim2NEW | ⊢ ¬ Lim 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 8530 | . 2 ⊢ 1o ∈ On | |
2 | nlimsuc 43344 | . . 3 ⊢ (1o ∈ On → ¬ Lim suc 1o) | |
3 | df-2o 8519 | . . . 4 ⊢ 2o = suc 1o | |
4 | limeq 6406 | . . . 4 ⊢ (2o = suc 1o → (Lim 2o ↔ Lim suc 1o)) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (Lim 2o ↔ Lim suc 1o) |
6 | 2, 5 | sylnibr 329 | . 2 ⊢ (1o ∈ On → ¬ Lim 2o) |
7 | 1, 6 | ax-mp 5 | 1 ⊢ ¬ Lim 2o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1537 ∈ wcel 2103 Oncon0 6394 Lim wlim 6395 suc csuc 6396 1oc1o 8511 2oc2o 8512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-tr 5287 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-1o 8518 df-2o 8519 |
This theorem is referenced by: (None) |
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