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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 4406 | . 2 ⊢ Ord ∅ | |
2 | 0ex 4145 | . . 3 ⊢ ∅ ∈ V | |
3 | 2 | elon 4389 | . 2 ⊢ (∅ ∈ On ↔ Ord ∅) |
4 | 1, 3 | mpbir 146 | 1 ⊢ ∅ ∈ On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 ∅c0 3437 Ord word 4377 Oncon0 4378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 ax-nul 4144 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-uni 3825 df-tr 4117 df-iord 4381 df-on 4383 |
This theorem is referenced by: inton 4408 onn0 4415 onm 4416 limon 4527 ordtriexmid 4535 ontriexmidim 4536 ordtri2orexmid 4537 onsucsssucexmid 4541 onsucelsucexmid 4544 ordsoexmid 4576 ordpwsucexmid 4584 ordtri2or2exmid 4585 ontri2orexmidim 4586 tfr0dm 6341 1on 6442 ordgt0ge1 6454 omv 6474 oa0 6476 om0 6477 oei0 6478 omcl 6480 omv2 6484 oaword1 6490 nna0r 6497 nnm0r 6498 card0 7205 |
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