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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 4410 | . 2 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) = ∅ ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐵))) | |
2 | ddif 4116 | . . 3 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵 | |
3 | 2 | sseq2i 3999 | . 2 ⊢ (𝐴 ⊆ (V ∖ (V ∖ 𝐵)) ↔ 𝐴 ⊆ 𝐵) |
4 | 1, 3 | bitr2i 278 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ (V ∖ 𝐵)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1536 Vcvv 3497 ∖ cdif 3936 ∩ cin 3938 ⊆ wss 3939 ∅c0 4294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-v 3499 df-dif 3942 df-in 3946 df-ss 3955 df-nul 4295 |
This theorem is referenced by: setind 9179 |
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