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Mirrors > Home > ILE Home > Th. List > sselii | GIF version |
Description: Membership inference from subclass relationship. (Contributed by NM, 31-May-1999.) |
Ref | Expression |
---|---|
sseli.1 | ⊢ 𝐴 ⊆ 𝐵 |
sselii.2 | ⊢ 𝐶 ∈ 𝐴 |
Ref | Expression |
---|---|
sselii | ⊢ 𝐶 ∈ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sselii.2 | . 2 ⊢ 𝐶 ∈ 𝐴 | |
2 | sseli.1 | . . 3 ⊢ 𝐴 ⊆ 𝐵 | |
3 | 2 | sseli 2941 | . 2 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵) |
4 | 1, 3 | ax-mp 7 | 1 ⊢ 𝐶 ∈ 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1393 ⊆ wss 2917 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 |
This theorem is referenced by: brtpos0 5867 ax1cn 6937 recni 7039 0xr 7072 nn0rei 8192 nnzi 8266 nn0zi 8267 pnfxr 8692 |
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