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Mirrors > Home > MPE Home > Th. List > 4onn | Structured version Visualization version GIF version |
Description: The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Ref | Expression |
---|---|
4onn | ⊢ 4o ∈ ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-4o 8508 | . 2 ⊢ 4o = suc 3o | |
2 | 3onn 8681 | . . 3 ⊢ 3o ∈ ω | |
3 | peano2 7913 | . . 3 ⊢ (3o ∈ ω → suc 3o ∈ ω) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ suc 3o ∈ ω |
5 | 1, 4 | eqeltri 2835 | 1 ⊢ 4o ∈ ω |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 suc csuc 6388 ωcom 7887 3oc3o 8500 4oc4o 8501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-om 7888 df-1o 8505 df-2o 8506 df-3o 8507 df-4o 8508 |
This theorem is referenced by: 4finon 43442 |
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