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| Description: Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.) | 
| Ref | Expression | 
|---|---|
| ssindif0 | ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | disj2 4457 | . 2 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵))) | |
| 2 | ddif 4140 | . . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵 | |
| 3 | 2 | sseq2i 4012 | . 2 ⊢ (𝐴 ⊆ (V ∖ (V ∖ 𝐵)) ↔ 𝐴 ⊆ 𝐵) | 
| 4 | 1, 3 | bitr2i 276 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1539 Vcvv 3479 ∖ cdif 3947 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-v 3481 df-dif 3953 df-in 3957 df-ss 3967 df-nul 4333 | 
| This theorem is referenced by: setind 9775 | 
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