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Mirrors > Home > MPE Home > Th. List > elnn | Structured version Visualization version GIF version |
Description: A member of a natural number is a natural number. (Contributed by NM, 21-Jun-1998.) |
Ref | Expression |
---|---|
elnn | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trom 7896 | . 2 ⊢ Tr ω | |
2 | trel 5274 | . 2 ⊢ (Tr ω → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ ω) → 𝐴 ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 Tr wtr 5265 ωcom 7887 |
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 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 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-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 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-om 7888 |
This theorem is referenced by: nnaordi 8655 nnmordi 8668 pssnn 9207 ssnnfi 9208 unfilem1 9341 unfilem2 9342 inf3lem5 9670 cantnflt 9710 cantnfp1lem3 9718 cantnflem1d 9726 cantnflem1 9727 cnfcomlem 9737 cnfcom 9738 ttrcltr 9754 ttrclselem2 9764 infpssrlem4 10344 axdc3lem2 10489 pwfseqlem3 10698 oldfi 27966 n0sbday 28369 pw2bday 28433 zs12bday 28439 bnj1098 34776 bnj517 34878 bnj594 34905 bnj1001 34952 bnj1118 34977 bnj1128 34983 bnj1145 34986 elhf2 36157 hfelhf 36163 |
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