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Mirrors > Home > ILE Home > Th. List > nn0ex | Unicode version |
Description: The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.) |
Ref | Expression |
---|---|
nn0ex |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-n0 8876 |
. 2
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2 | nnex 8630 |
. . 3
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3 | c0ex 7678 |
. . . 4
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4 | 3 | snex 4067 |
. . 3
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5 | 2, 4 | unex 4320 |
. 2
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6 | 1, 5 | eqeltri 2185 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-i2m1 7644 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-uni 3701 df-int 3736 df-inn 8625 df-n0 8876 |
This theorem is referenced by: nn0ennn 10093 nnenom 10094 uzennn 10096 expcnvap0 11157 expcnvre 11158 expcnv 11159 geolim 11166 mertenslem2 11191 eftlub 11241 znnen 11750 |
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