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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 7562 | . 2 ⊢ 3𝑜 = suc 2𝑜 | |
2 | 2onn 7720 | . . 3 ⊢ 2𝑜 ∈ ω | |
3 | peano2 7086 | . . 3 ⊢ (2𝑜 ∈ ω → suc 2𝑜 ∈ ω) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ suc 2𝑜 ∈ ω |
5 | 1, 4 | eqeltri 2697 | 1 ⊢ 3𝑜 ∈ ω |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 suc csuc 5725 ωcom 7065 2𝑜c2o 7554 3𝑜c3o 7555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-om 7066 df-1o 7560 df-2o 7561 df-3o 7562 |
This theorem is referenced by: 4onn 7722 en4 8198 hash4 13195 hash3tr 13272 |
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