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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 3901 | . 2 ⊢ 𝐴 ⊆ V | |
2 | reldisj 4341 | . 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 209 = wceq 1542 Vcvv 3398 ∖ cdif 3840 ∩ cin 3842 ⊆ wss 3843 ∅c0 4211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-v 3400 df-dif 3846 df-in 3850 df-ss 3860 df-nul 4212 |
This theorem is referenced by: ssindif0 4353 intirr 5952 setsres 16628 setscom 16630 f1omvdco3 18695 psgnunilem5 18740 opsrtoslem2 20867 clsconn 22181 cldsubg 22862 uniinn0 30464 imadifxp 30514 |
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