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Mirrors > Home > ILE Home > Th. List > nlim0 | GIF version |
Description: The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
nlim0 | ⊢ ¬ Lim ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3398 | . . 3 ⊢ ¬ ∅ ∈ ∅ | |
2 | simp2 983 | . . 3 ⊢ ((Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅) → ∅ ∈ ∅) | |
3 | 1, 2 | mto 652 | . 2 ⊢ ¬ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅) |
4 | dflim2 4330 | . 2 ⊢ (Lim ∅ ↔ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅)) | |
5 | 3, 4 | mtbir 661 | 1 ⊢ ¬ Lim ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ w3a 963 = wceq 1335 ∈ wcel 2128 ∅c0 3394 ∪ cuni 3772 Ord word 4322 Lim wlim 4324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-dif 3104 df-nul 3395 df-ilim 4329 |
This theorem is referenced by: (None) |
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