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Mirrors > Home > MPE Home > Th. List > ssindif0 | Structured version Visualization version GIF version |
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 3976 | . 2 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵))) | |
2 | ddif 3704 | . . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵 | |
3 | 2 | sseq2i 3593 | . 2 ⊢ (𝐴 ⊆ (V ∖ (V ∖ 𝐵)) ↔ 𝐴 ⊆ 𝐵) |
4 | 1, 3 | bitr2i 264 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 Vcvv 3173 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 |
This theorem is referenced by: setind 8493 |
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