Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3418 | . . 3 ⊢ ¬ ∅ ∈ ∅ | |
2 | simp2 993 | . . 3 ⊢ ((Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅) → ∅ ∈ ∅) | |
3 | 1, 2 | mto 657 | . 2 ⊢ ¬ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅) |
4 | dflim2 4355 | . 2 ⊢ (Lim ∅ ↔ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅)) | |
5 | 3, 4 | mtbir 666 | 1 ⊢ ¬ Lim ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ∅c0 3414 ∪ cuni 3796 Ord word 4347 Lim wlim 4349 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-nul 3415 df-ilim 4354 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |