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Mirrors > Home > ILE Home > Th. List > 4onn | GIF version |
Description: The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Ref | Expression |
---|---|
4onn | ⊢ 4o ∈ ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-4o 6246 | . 2 ⊢ 4o = suc 3o | |
2 | 3onn 6348 | . . 3 ⊢ 3o ∈ ω | |
3 | peano2 4447 | . . 3 ⊢ (3o ∈ ω → suc 3o ∈ ω) | |
4 | 2, 3 | ax-mp 7 | . 2 ⊢ suc 3o ∈ ω |
5 | 1, 4 | eqeltri 2172 | 1 ⊢ 4o ∈ ω |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1448 suc csuc 4225 ωcom 4442 3oc3o 6238 4oc4o 6239 |
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-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-uni 3684 df-int 3719 df-suc 4231 df-iom 4443 df-1o 6243 df-2o 6244 df-3o 6245 df-4o 6246 |
This theorem is referenced by: (None) |
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