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Mirrors > Home > ILE Home > Th. List > dfss | GIF version |
Description: Variant of subclass definition df-ss 2931. (Contributed by NM, 3-Sep-2004.) |
Ref | Expression |
---|---|
dfss | ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ss 2931 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
2 | eqcom 2042 | . 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 ↔ 𝐴 = (𝐴 ∩ 𝐵)) | |
3 | 1, 2 | bitri 173 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1243 ∩ cin 2916 ⊆ 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-gen 1338 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 df-ss 2931 |
This theorem is referenced by: dfss2 2934 onelini 4167 cnvcnv 4773 funimass1 4976 dmaddpi 6423 dmmulpi 6424 |
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