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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 3988 | . 2 ⊢ 𝐴 ⊆ V | |
2 | reldisj 4398 | . 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 207 = wceq 1528 Vcvv 3492 ∖ cdif 3930 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-v 3494 df-dif 3936 df-in 3940 df-ss 3949 df-nul 4289 |
This theorem is referenced by: ssindif0 4409 intirr 5971 setsres 16513 setscom 16515 f1omvdco3 18506 psgnunilem5 18551 opsrtoslem2 20193 clsconn 21966 cldsubg 22646 uniinn0 30229 imadifxp 30279 |
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