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Mirrors > Home > ILE Home > Th. List > 2onn | GIF version |
Description: The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.) |
Ref | Expression |
---|---|
2onn | ⊢ 2𝑜 ∈ ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2o 6002 | . 2 ⊢ 2𝑜 = suc 1𝑜 | |
2 | 1onn 6093 | . . 3 ⊢ 1𝑜 ∈ ω | |
3 | peano2 4318 | . . 3 ⊢ (1𝑜 ∈ ω → suc 1𝑜 ∈ ω) | |
4 | 2, 3 | ax-mp 7 | . 2 ⊢ suc 1𝑜 ∈ ω |
5 | 1, 4 | eqeltri 2110 | 1 ⊢ 2𝑜 ∈ ω |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1393 suc csuc 4102 ωcom 4313 1𝑜c1o 5994 2𝑜c2o 5995 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-int 3616 df-suc 4108 df-iom 4314 df-1o 6001 df-2o 6002 |
This theorem is referenced by: 3onn 6095 nn2m 6099 prarloclemarch2 6517 nq02m 6563 prarloclemlt 6591 prarloclemlo 6592 prarloclem3 6595 prarloclemn 6597 prarloclem5 6598 prarloclemcalc 6600 |
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