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Mirrors > Home > ILE Home > Th. List > 4onn | 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 6423 | . 2 ⊢ 4o = suc 3o | |
2 | 3onn 6526 | . . 3 ⊢ 3o ∈ ω | |
3 | peano2 4596 | . . 3 ⊢ (3o ∈ ω → suc 3o ∈ ω) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ suc 3o ∈ ω |
5 | 1, 4 | eqeltri 2250 | 1 ⊢ 4o ∈ ω |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 suc csuc 4367 ωcom 4591 3oc3o 6415 4oc4o 6416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-uni 3812 df-int 3847 df-suc 4373 df-iom 4592 df-1o 6420 df-2o 6421 df-3o 6422 df-4o 6423 |
This theorem is referenced by: (None) |
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