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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 5268 | . 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 2108 Tr wtr 5259 ωcom 7887 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-lim 6389 df-om 7888 |
| This theorem is referenced by: nnaordi 8656 nnmordi 8669 pssnn 9208 ssnnfi 9209 unfilem1 9343 unfilem2 9344 inf3lem5 9672 cantnflt 9712 cantnfp1lem3 9720 cantnflem1d 9728 cantnflem1 9729 cnfcomlem 9739 cnfcom 9740 ttrcltr 9756 ttrclselem2 9766 infpssrlem4 10346 axdc3lem2 10491 pwfseqlem3 10700 oldfi 27951 n0sbday 28354 pw2bday 28418 zs12bday 28424 bnj1098 34797 bnj517 34899 bnj594 34926 bnj1001 34973 bnj1118 34998 bnj1128 35004 bnj1145 35007 elhf2 36176 hfelhf 36182 |
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