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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 6282 | . 2 ⊢ 2o = suc 1o | |
2 | 1onn 6384 | . . 3 ⊢ 1o ∈ ω | |
3 | peano2 4479 | . . 3 ⊢ (1o ∈ ω → suc 1o ∈ ω) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ suc 1o ∈ ω |
5 | 1, 4 | eqeltri 2190 | 1 ⊢ 2o ∈ ω |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 suc csuc 4257 ωcom 4474 1oc1o 6274 2oc2o 6275 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-int 3742 df-suc 4263 df-iom 4475 df-1o 6281 df-2o 6282 |
This theorem is referenced by: 3onn 6386 nn2m 6390 isomnimap 6977 enomnilem 6978 fodjuf 6985 infnninf 6990 nnnninf 6991 ismkvmap 6996 ismkvnex 6997 exmidonfinlem 7017 exmidfodomrlemr 7026 exmidfodomrlemrALT 7027 prarloclemarch2 7195 nq02m 7241 prarloclemlt 7269 prarloclemlo 7270 prarloclem3 7273 prarloclemn 7275 prarloclem5 7276 prarloclemcalc 7278 hash3 10527 unct 11881 pwle2 13120 pwf1oexmid 13121 subctctexmid 13123 0nninf 13124 nnsf 13126 nninfex 13132 nninfsellemdc 13133 nninfself 13136 nninffeq 13143 isomninnlem 13152 |
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