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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 6316 | . 2 ⊢ 4o = suc 3o | |
2 | 3onn 6418 | . . 3 ⊢ 3o ∈ ω | |
3 | peano2 4509 | . . 3 ⊢ (3o ∈ ω → suc 3o ∈ ω) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ suc 3o ∈ ω |
5 | 1, 4 | eqeltri 2212 | 1 ⊢ 4o ∈ ω |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 suc csuc 4287 ωcom 4504 3oc3o 6308 4oc4o 6309 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-suc 4293 df-iom 4505 df-1o 6313 df-2o 6314 df-3o 6315 df-4o 6316 |
This theorem is referenced by: (None) |
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