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Mirrors > Home > ILE Home > Th. List > sselda | GIF version |
Description: Membership deduction from subclass relationship. (Contributed by NM, 26-Jun-2014.) |
Ref | Expression |
---|---|
sseld.1 | ⊢ (φ → A ⊆ B) |
Ref | Expression |
---|---|
sselda | ⊢ ((φ ∧ 𝐶 ∈ A) → 𝐶 ∈ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseld.1 | . . 3 ⊢ (φ → A ⊆ B) | |
2 | 1 | sseld 2938 | . 2 ⊢ (φ → (𝐶 ∈ A → 𝐶 ∈ B)) |
3 | 2 | imp 115 | 1 ⊢ ((φ ∧ 𝐶 ∈ A) → 𝐶 ∈ B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 ⊆ wss 2911 |
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 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-in 2918 df-ss 2925 |
This theorem is referenced by: elrel 4385 ffvresb 5271 1stdm 5750 tfrlem1 5864 tfrlemiubacc 5885 erinxp 6116 fundmen 6222 elprnql 6464 elprnqu 6465 un0addcl 7991 un0mulcl 7992 icoshftf1o 8629 elfzom1elfzo 8829 zpnn0elfzo 8833 iseqfveq 8907 |
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