Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 2onn | Structured version Visualization version 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 8092 | . 2 ⊢ 2o = suc 1o | |
2 | 1onn 8254 | . . 3 ⊢ 1o ∈ ω | |
3 | peano2 7591 | . . 3 ⊢ (1o ∈ ω → suc 1o ∈ ω) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ suc 1o ∈ ω |
5 | 1, 4 | eqeltri 2906 | 1 ⊢ 2o ∈ ω |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 suc csuc 6186 ωcom 7569 1oc1o 8084 2oc2o 8085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-om 7570 df-1o 8091 df-2o 8092 |
This theorem is referenced by: 3onn 8256 nn2m 8266 nnneo 8267 nneob 8268 omopthlem1 8271 omopthlem2 8272 pwen 8678 en3 8743 en2eqpr 9421 en2eleq 9422 unctb 9615 infdjuabs 9616 ackbij1lem5 9634 sdom2en01 9712 fin56 9803 fin67 9805 fin1a2lem4 9813 alephexp1 9989 pwcfsdom 9993 alephom 9995 canthp1lem2 10063 pwxpndom2 10075 hash3 13755 hash2pr 13815 pr2pwpr 13825 rpnnen 15568 rexpen 15569 xpsfrnel 16823 xpscf 16826 symggen 18527 psgnunilem1 18550 simpgnsgd 19151 znfld 20635 hauspwdom 22037 xpsmet 22919 xpsxms 23071 xpsms 23072 unidifsnel 30222 unidifsnne 30223 sat1el2xp 32523 ex-sategoelelomsuc 32570 ex-sategoelel12 32571 1oequni2o 34531 finxpreclem4 34557 finxp3o 34563 wepwso 39521 frlmpwfi 39576 |
Copyright terms: Public domain | W3C validator |