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Mirrors > Home > MPE Home > Th. List > disj2 | Structured version Visualization version GIF version |
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 3991 | . 2 ⊢ 𝐴 ⊆ V | |
2 | reldisj 4402 | . 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 208 = wceq 1533 Vcvv 3495 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3497 df-dif 3939 df-in 3943 df-ss 3952 df-nul 4292 |
This theorem is referenced by: ssindif0 4413 intirr 5973 setsres 16519 setscom 16521 f1omvdco3 18571 psgnunilem5 18616 opsrtoslem2 20259 clsconn 22032 cldsubg 22713 uniinn0 30296 imadifxp 30345 |
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