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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 8474 | . 2 ⊢ 3o = suc 2o | |
2 | 2onn 8647 | . . 3 ⊢ 2o ∈ ω | |
3 | peano2 7885 | . . 3 ⊢ (2o ∈ ω → suc 2o ∈ ω) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ suc 2o ∈ ω |
5 | 1, 4 | eqeltri 2828 | 1 ⊢ 3o ∈ ω |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 suc csuc 6366 ωcom 7859 2oc2o 8466 3oc3o 8467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-om 7860 df-1o 8472 df-2o 8473 df-3o 8474 |
This theorem is referenced by: 4onn 8650 hash4 14374 hash3tr 14458 oenord1ex 42380 3finon 42517 |
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