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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 4227 | . 2 ⊢ Ord ∅ | |
2 | 0ex 3972 | . . 3 ⊢ ∅ ∈ V | |
3 | 2 | elon 4210 | . 2 ⊢ (∅ ∈ On ↔ Ord ∅) |
4 | 1, 3 | mpbir 145 | 1 ⊢ ∅ ∈ On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1439 ∅c0 3287 Ord word 4198 Oncon0 4199 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-nul 3971 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-dif 3002 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-uni 3660 df-tr 3943 df-iord 4202 df-on 4204 |
This theorem is referenced by: inton 4229 onn0 4236 onm 4237 limon 4343 ordtriexmid 4351 ordtri2orexmid 4352 onsucsssucexmid 4356 onsucelsucexmid 4359 ordsoexmid 4391 ordpwsucexmid 4399 ordtri2or2exmid 4400 tfr0dm 6101 1on 6202 ordgt0ge1 6213 omv 6230 oa0 6232 om0 6233 oei0 6234 omcl 6236 omv2 6240 oaword1 6246 nna0r 6253 nnm0r 6254 card0 6870 |
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