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Mirrors > Home > MPE Home > Th. List > disj4 | Structured version Visualization version GIF version |
Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 21-Mar-2004.) |
Ref | Expression |
---|---|
disj4 | ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ¬ (𝐴 ∖ 𝐵) ⊊ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disj3 4405 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵)) | |
2 | eqcom 2830 | . 2 ⊢ (𝐴 = (𝐴 ∖ 𝐵) ↔ (𝐴 ∖ 𝐵) = 𝐴) | |
3 | difss 4110 | . . . 4 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
4 | dfpss2 4064 | . . . 4 ⊢ ((𝐴 ∖ 𝐵) ⊊ 𝐴 ↔ ((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ ¬ (𝐴 ∖ 𝐵) = 𝐴)) | |
5 | 3, 4 | mpbiran 707 | . . 3 ⊢ ((𝐴 ∖ 𝐵) ⊊ 𝐴 ↔ ¬ (𝐴 ∖ 𝐵) = 𝐴) |
6 | 5 | con2bii 360 | . 2 ⊢ ((𝐴 ∖ 𝐵) = 𝐴 ↔ ¬ (𝐴 ∖ 𝐵) ⊊ 𝐴) |
7 | 1, 2, 6 | 3bitri 299 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ¬ (𝐴 ∖ 𝐵) ⊊ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1537 ∖ cdif 3935 ∩ cin 3937 ⊆ wss 3938 ⊊ wpss 3939 ∅c0 4293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 |
This theorem is referenced by: marypha1lem 8899 infeq5i 9101 wilthlem2 25648 topdifinffinlem 34630 |
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