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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 3059 | . 2 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵) |
4 | 1, 3 | ax-mp 7 | 1 ⊢ 𝐶 ∈ 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1463 ⊆ wss 3037 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-11 1467 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-in 3043 df-ss 3050 |
This theorem is referenced by: brtpos0 6103 ax1cn 7596 recni 7702 0xr 7736 pnfxr 7742 nn0rei 8892 0xnn0 8950 nnzi 8979 nn0zi 8980 |
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