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| Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.) | 
| Ref | Expression | 
|---|---|
| disj2 | ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssv 4008 | . 2 ⊢ 𝐴 ⊆ V | |
| 2 | reldisj 4453 | . 2 ⊢ (𝐴 ⊆ V → ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 Vcvv 3480 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 | 
| This theorem is referenced by: ssindif0 4464 intirr 6138 setsres 17215 setscom 17217 f1omvdco3 19467 psgnunilem5 19512 opsrtoslem2 22080 clsconn 23438 cldsubg 24119 uniinn0 32563 imadifxp 32614 | 
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