Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > 0elon | GIF version |
Description: The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.) |
Ref | Expression |
---|---|
0elon | ⊢ ∅ ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ord0 4351 | . 2 ⊢ Ord ∅ | |
2 | 0ex 4091 | . . 3 ⊢ ∅ ∈ V | |
3 | 2 | elon 4334 | . 2 ⊢ (∅ ∈ On ↔ Ord ∅) |
4 | 1, 3 | mpbir 145 | 1 ⊢ ∅ ∈ On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2128 ∅c0 3394 Ord word 4322 Oncon0 4323 |
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 ax-nul 4090 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-uni 3773 df-tr 4063 df-iord 4326 df-on 4328 |
This theorem is referenced by: inton 4353 onn0 4360 onm 4361 limon 4472 ordtriexmid 4480 ontriexmidim 4481 ordtri2orexmid 4482 onsucsssucexmid 4486 onsucelsucexmid 4489 ordsoexmid 4521 ordpwsucexmid 4529 ordtri2or2exmid 4530 ontri2orexmidim 4531 tfr0dm 6269 1on 6370 ordgt0ge1 6382 omv 6402 oa0 6404 om0 6405 oei0 6406 omcl 6408 omv2 6412 oaword1 6418 nna0r 6425 nnm0r 6426 card0 7123 |
Copyright terms: Public domain | W3C validator |