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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 3021 | . 2 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵) |
4 | 1, 3 | ax-mp 7 | 1 ⊢ 𝐶 ∈ 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1438 ⊆ wss 2999 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-11 1442 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-in 3005 df-ss 3012 |
This theorem is referenced by: brtpos0 6017 ax1cn 7398 recni 7500 0xr 7534 pnfxr 7540 nn0rei 8684 0xnn0 8742 nnzi 8771 nn0zi 8772 |
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