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Mirrors > Home > ILE Home > Th. List > dfss | GIF version |
Description: Variant of subclass definition df-ss 3084. (Contributed by NM, 3-Sep-2004.) |
Ref | Expression |
---|---|
dfss | ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ss 3084 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
2 | eqcom 2141 | . 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 ↔ 𝐴 = (𝐴 ∩ 𝐵)) | |
3 | 1, 2 | bitri 183 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∩ cin 3070 ⊆ wss 3071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-cleq 2132 df-ss 3084 |
This theorem is referenced by: dfss2 3086 onelini 4352 cnvcnv 4991 funimass1 5200 sbthlemi5 6849 dmaddpi 7133 dmmulpi 7134 tgioo 12715 |
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