![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nlim2 | Structured version Visualization version GIF version |
Description: 2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) |
Ref | Expression |
---|---|
nlim2 | ⊢ ¬ Lim 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1oex 8418 | . . . . . . . . 9 ⊢ 1o ∈ V | |
2 | 1 | prid2 4722 | . . . . . . . 8 ⊢ 1o ∈ {∅, 1o} |
3 | df2o3 8416 | . . . . . . . 8 ⊢ 2o = {∅, 1o} | |
4 | 2, 3 | eleqtrri 2837 | . . . . . . 7 ⊢ 1o ∈ 2o |
5 | 1on 8420 | . . . . . . . . 9 ⊢ 1o ∈ On | |
6 | 5 | onirri 6427 | . . . . . . . 8 ⊢ ¬ 1o ∈ 1o |
7 | eleq2 2826 | . . . . . . . 8 ⊢ (2o = 1o → (1o ∈ 2o ↔ 1o ∈ 1o)) | |
8 | 6, 7 | mtbiri 326 | . . . . . . 7 ⊢ (2o = 1o → ¬ 1o ∈ 2o) |
9 | 4, 8 | mt2 199 | . . . . . 6 ⊢ ¬ 2o = 1o |
10 | 9 | neir 2944 | . . . . 5 ⊢ 2o ≠ 1o |
11 | 3 | unieqi 4876 | . . . . . 6 ⊢ ∪ 2o = ∪ {∅, 1o} |
12 | 0ex 5262 | . . . . . . 7 ⊢ ∅ ∈ V | |
13 | 12, 1 | unipr 4881 | . . . . . 6 ⊢ ∪ {∅, 1o} = (∅ ∪ 1o) |
14 | 0un 4350 | . . . . . 6 ⊢ (∅ ∪ 1o) = 1o | |
15 | 11, 13, 14 | 3eqtri 2768 | . . . . 5 ⊢ ∪ 2o = 1o |
16 | 10, 15 | neeqtrri 3015 | . . . 4 ⊢ 2o ≠ ∪ 2o |
17 | 16 | neii 2943 | . . 3 ⊢ ¬ 2o = ∪ 2o |
18 | simp3 1138 | . . 3 ⊢ ((Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o) → 2o = ∪ 2o) | |
19 | 17, 18 | mto 196 | . 2 ⊢ ¬ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o) |
20 | df-lim 6320 | . 2 ⊢ (Lim 2o ↔ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o)) | |
21 | 19, 20 | mtbir 322 | 1 ⊢ ¬ Lim 2o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∪ cun 3906 ∅c0 4280 {cpr 4586 ∪ cuni 4863 Ord word 6314 Lim wlim 6316 1oc1o 8401 2oc2o 8402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-tr 5221 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-1o 8408 df-2o 8409 |
This theorem is referenced by: 2ellim 8441 |
Copyright terms: Public domain | W3C validator |