Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4368 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵)) | |
2 | eqcom 2744 | . 2 ⊢ (𝐴 = (𝐴 ∖ 𝐵) ↔ (𝐴 ∖ 𝐵) = 𝐴) | |
3 | difss 4046 | . . . 4 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
4 | dfpss2 4000 | . . . 4 ⊢ ((𝐴 ∖ 𝐵) ⊊ 𝐴 ↔ ((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ ¬ (𝐴 ∖ 𝐵) = 𝐴)) | |
5 | 3, 4 | mpbiran 709 | . . 3 ⊢ ((𝐴 ∖ 𝐵) ⊊ 𝐴 ↔ ¬ (𝐴 ∖ 𝐵) = 𝐴) |
6 | 5 | con2bii 361 | . 2 ⊢ ((𝐴 ∖ 𝐵) = 𝐴 ↔ ¬ (𝐴 ∖ 𝐵) ⊊ 𝐴) |
7 | 1, 2, 6 | 3bitri 300 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ¬ (𝐴 ∖ 𝐵) ⊊ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1543 ∖ cdif 3863 ∩ cin 3865 ⊆ wss 3866 ⊊ wpss 3867 ∅c0 4237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-v 3410 df-dif 3869 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 |
This theorem is referenced by: marypha1lem 9049 infeq5i 9251 wilthlem2 25951 topdifinffinlem 35255 |
Copyright terms: Public domain | W3C validator |