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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 4376 | . 2 ⊢ Ord ∅ | |
2 | 0ex 4116 | . . 3 ⊢ ∅ ∈ V | |
3 | 2 | elon 4359 | . 2 ⊢ (∅ ∈ On ↔ Ord ∅) |
4 | 1, 3 | mpbir 145 | 1 ⊢ ∅ ∈ On |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 ∅c0 3414 Ord word 4347 Oncon0 4348 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-nul 4115 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-uni 3797 df-tr 4088 df-iord 4351 df-on 4353 |
This theorem is referenced by: inton 4378 onn0 4385 onm 4386 limon 4497 ordtriexmid 4505 ontriexmidim 4506 ordtri2orexmid 4507 onsucsssucexmid 4511 onsucelsucexmid 4514 ordsoexmid 4546 ordpwsucexmid 4554 ordtri2or2exmid 4555 ontri2orexmidim 4556 tfr0dm 6301 1on 6402 ordgt0ge1 6414 omv 6434 oa0 6436 om0 6437 oei0 6438 omcl 6440 omv2 6444 oaword1 6450 nna0r 6457 nnm0r 6458 card0 7165 |
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