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Mirrors > Home > ILE Home > Th. List > 3onn | GIF version |
Description: The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Ref | Expression |
---|---|
3onn | ⊢ 3o ∈ ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3o 6458 | . 2 ⊢ 3o = suc 2o | |
2 | 2onn 6561 | . . 3 ⊢ 2o ∈ ω | |
3 | peano2 4619 | . . 3 ⊢ (2o ∈ ω → suc 2o ∈ ω) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ suc 2o ∈ ω |
5 | 1, 4 | eqeltri 2262 | 1 ⊢ 3o ∈ ω |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 suc csuc 4390 ωcom 4614 2oc2o 6450 3oc3o 6451 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-nul 4151 ax-pow 4199 ax-pr 4234 ax-un 4458 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2758 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-nul 3443 df-pw 3599 df-sn 3620 df-pr 3621 df-uni 3832 df-int 3867 df-suc 4396 df-iom 4615 df-1o 6456 df-2o 6457 df-3o 6458 |
This theorem is referenced by: 4onn 6563 hash4 10859 |
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