Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > limon | GIF version |
Description: The class of ordinal numbers is a limit ordinal. (Contributed by NM, 24-Mar-1995.) |
Ref | Expression |
---|---|
limon | ⊢ Lim On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordon 4402 | . 2 ⊢ Ord On | |
2 | 0elon 4314 | . 2 ⊢ ∅ ∈ On | |
3 | unon 4427 | . . 3 ⊢ ∪ On = On | |
4 | 3 | eqcomi 2143 | . 2 ⊢ On = ∪ On |
5 | dflim2 4292 | . 2 ⊢ (Lim On ↔ (Ord On ∧ ∅ ∈ On ∧ On = ∪ On)) | |
6 | 1, 2, 4, 5 | mpbir3an 1163 | 1 ⊢ Lim On |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 ∅c0 3363 ∪ cuni 3736 Ord word 4284 Oncon0 4285 Lim wlim 4286 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-tr 4027 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |