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Mirrors > Home > ILE Home > Th. List > sseldd | GIF version |
Description: Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004.) |
Ref | Expression |
---|---|
sseld.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
sseldd.2 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
Ref | Expression |
---|---|
sseldd | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseldd.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
2 | sseld.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
3 | 2 | sseld 2944 | . 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)) |
4 | 1, 3 | mpd 13 | 1 ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ 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: frirrg 4087 ordtri2or2exmid 4296 riotass 5495 eroveu 6197 eroprf 6199 findcard2d 6348 findcard2sd 6349 nnppipi 6441 archnqq 6515 prarloclemlt 6591 fzssp1 8930 elfzoelz 9004 fzofzp1 9083 fzostep1 9093 isermono 9237 |
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