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| 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 8516 | . . . . . . . . 9 ⊢ 1o ∈ V | |
| 2 | 1 | prid2 4763 | . . . . . . . 8 ⊢ 1o ∈ {∅, 1o} |
| 3 | df2o3 8514 | . . . . . . . 8 ⊢ 2o = {∅, 1o} | |
| 4 | 2, 3 | eleqtrri 2840 | . . . . . . 7 ⊢ 1o ∈ 2o |
| 5 | 1on 8518 | . . . . . . . . 9 ⊢ 1o ∈ On | |
| 6 | 5 | onirri 6497 | . . . . . . . 8 ⊢ ¬ 1o ∈ 1o |
| 7 | eleq2 2830 | . . . . . . . 8 ⊢ (2o = 1o → (1o ∈ 2o ↔ 1o ∈ 1o)) | |
| 8 | 6, 7 | mtbiri 327 | . . . . . . 7 ⊢ (2o = 1o → ¬ 1o ∈ 2o) |
| 9 | 4, 8 | mt2 200 | . . . . . 6 ⊢ ¬ 2o = 1o |
| 10 | 9 | neir 2943 | . . . . 5 ⊢ 2o ≠ 1o |
| 11 | 3 | unieqi 4919 | . . . . . 6 ⊢ ∪ 2o = ∪ {∅, 1o} |
| 12 | 0ex 5307 | . . . . . . 7 ⊢ ∅ ∈ V | |
| 13 | 12, 1 | unipr 4924 | . . . . . 6 ⊢ ∪ {∅, 1o} = (∅ ∪ 1o) |
| 14 | 0un 4396 | . . . . . 6 ⊢ (∅ ∪ 1o) = 1o | |
| 15 | 11, 13, 14 | 3eqtri 2769 | . . . . 5 ⊢ ∪ 2o = 1o |
| 16 | 10, 15 | neeqtrri 3014 | . . . 4 ⊢ 2o ≠ ∪ 2o |
| 17 | 16 | neii 2942 | . . 3 ⊢ ¬ 2o = ∪ 2o |
| 18 | simp3 1139 | . . 3 ⊢ ((Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o) → 2o = ∪ 2o) | |
| 19 | 17, 18 | mto 197 | . 2 ⊢ ¬ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o) |
| 20 | df-lim 6389 | . 2 ⊢ (Lim 2o ↔ (Ord 2o ∧ 2o ≠ ∅ ∧ 2o = ∪ 2o)) | |
| 21 | 19, 20 | mtbir 323 | 1 ⊢ ¬ Lim 2o |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∪ cun 3949 ∅c0 4333 {cpr 4628 ∪ cuni 4907 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: 2ellim 8537 |
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