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Mirrors > Home > ILE Home > Th. List > peano2b | Unicode version |
Description: A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
Ref | Expression |
---|---|
peano2b |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2 4410 |
. 2
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2 | elex 2630 |
. . . . 5
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3 | sucexb 4314 |
. . . . 5
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4 | 2, 3 | sylibr 132 |
. . . 4
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5 | sucidg 4243 |
. . . 4
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6 | 4, 5 | syl 14 |
. . 3
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7 | elnn 4420 |
. . 3
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8 | 6, 7 | mpancom 413 |
. 2
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9 | 1, 8 | impbii 124 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-iinf 4403 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-uni 3654 df-int 3689 df-suc 4198 df-iom 4406 |
This theorem is referenced by: nnmsucr 6249 nnpredcl 11890 |
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