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Mirrors > Home > MPE Home > Th. List > 3onn | Structured version Visualization version 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 7905 | . 2 ⊢ 3o = suc 2o | |
2 | 2onn 8065 | . . 3 ⊢ 2o ∈ ω | |
3 | peano2 7415 | . . 3 ⊢ (2o ∈ ω → suc 2o ∈ ω) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ suc 2o ∈ ω |
5 | 1, 4 | eqeltri 2855 | 1 ⊢ 3o ∈ ω |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2051 suc csuc 6028 ωcom 7394 2oc2o 7897 3oc3o 7898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-tr 5027 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-om 7395 df-1o 7903 df-2o 7904 df-3o 7905 |
This theorem is referenced by: 4onn 8067 en4 8549 hash4 13579 hash3tr 13657 |
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