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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 7606 | . 2 ⊢ 2𝑜 = suc 1𝑜 | |
2 | 1onn 7764 | . . 3 ⊢ 1𝑜 ∈ ω | |
3 | peano2 7128 | . . 3 ⊢ (1𝑜 ∈ ω → suc 1𝑜 ∈ ω) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ suc 1𝑜 ∈ ω |
5 | 1, 4 | eqeltri 2726 | 1 ⊢ 2𝑜 ∈ ω |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2030 suc csuc 5763 ωcom 7107 1𝑜c1o 7598 2𝑜c2o 7599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-tr 4786 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-om 7108 df-1o 7605 df-2o 7606 |
This theorem is referenced by: 3onn 7766 nn2m 7775 nnneo 7776 nneob 7777 omopthlem1 7780 omopthlem2 7781 pwen 8174 en3 8238 en2eqpr 8868 en2eleq 8869 unctb 9065 infcdaabs 9066 ackbij1lem5 9084 sdom2en01 9162 fin56 9253 fin67 9255 fin1a2lem4 9263 alephexp1 9439 pwcfsdom 9443 alephom 9445 canthp1lem2 9513 pwxpndom2 9525 hash3 13232 hash2pr 13289 pr2pwpr 13299 rpnnen 15000 rexpen 15001 xpsfrnel 16270 symggen 17936 psgnunilem1 17959 znfld 19957 hauspwdom 21352 xpsmet 22234 xpsxms 22386 xpsms 22387 1oequni2o 33346 finxpreclem4 33361 finxp3o 33367 wepwso 37930 frlmpwfi 37985 |
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