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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 8201 | . 2 ⊢ 3o = suc 2o | |
2 | 2onn 8365 | . . 3 ⊢ 2o ∈ ω | |
3 | peano2 7665 | . . 3 ⊢ (2o ∈ ω → suc 2o ∈ ω) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ suc 2o ∈ ω |
5 | 1, 4 | eqeltri 2834 | 1 ⊢ 3o ∈ ω |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 suc csuc 6212 ωcom 7641 2oc2o 8193 3oc3o 8194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pr 5319 ax-un 7520 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2940 df-ral 3063 df-rex 3064 df-rab 3067 df-v 3407 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-br 5051 df-opab 5113 df-tr 5159 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-om 7642 df-1o 8199 df-2o 8200 df-3o 8201 |
This theorem is referenced by: 4onn 8367 en4 8909 hash4 13971 hash3tr 14053 |
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