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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 3133 | . 2 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ 𝐶 ∈ 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2135 ⊆ wss 3111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-in 3117 df-ss 3124 |
This theorem is referenced by: brtpos0 6211 ax1cn 7793 recni 7902 0xr 7936 pnfxr 7942 nn0rei 9116 0xnn0 9174 nnzi 9203 nn0zi 9204 |
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