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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 4321 | . 2 ⊢ Ord ∅ | |
2 | 0ex 4063 | . . 3 ⊢ ∅ ∈ V | |
3 | 2 | elon 4304 | . 2 ⊢ (∅ ∈ On ↔ Ord ∅) |
4 | 1, 3 | mpbir 145 | 1 ⊢ ∅ ∈ On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 ∅c0 3368 Ord word 4292 Oncon0 4293 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-nul 4062 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-uni 3745 df-tr 4035 df-iord 4296 df-on 4298 |
This theorem is referenced by: inton 4323 onn0 4330 onm 4331 limon 4437 ordtriexmid 4445 ordtri2orexmid 4446 onsucsssucexmid 4450 onsucelsucexmid 4453 ordsoexmid 4485 ordpwsucexmid 4493 ordtri2or2exmid 4494 tfr0dm 6227 1on 6328 ordgt0ge1 6340 omv 6359 oa0 6361 om0 6362 oei0 6363 omcl 6365 omv2 6369 oaword1 6375 nna0r 6382 nnm0r 6383 card0 7061 |
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