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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 7449 | . 2 ⊢ 3𝑜 = suc 2𝑜 | |
2 | 2onn 7607 | . . 3 ⊢ 2𝑜 ∈ ω | |
3 | peano2 6978 | . . 3 ⊢ (2𝑜 ∈ ω → suc 2𝑜 ∈ ω) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ suc 2𝑜 ∈ ω |
5 | 1, 4 | eqeltri 2684 | 1 ⊢ 3𝑜 ∈ ω |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 suc csuc 5642 ωcom 6957 2𝑜c2o 7441 3𝑜c3o 7442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-om 6958 df-1o 7447 df-2o 7448 df-3o 7449 |
This theorem is referenced by: 4onn 7609 en4 8083 hash4 13056 hash3tr 13127 |
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