![]() |
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 4164 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵)) | |
2 | eqcom 2767 | . 2 ⊢ (𝐴 = (𝐴 ∖ 𝐵) ↔ (𝐴 ∖ 𝐵) = 𝐴) | |
3 | difss 3880 | . . . 4 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
4 | dfpss2 3834 | . . . 4 ⊢ ((𝐴 ∖ 𝐵) ⊊ 𝐴 ↔ ((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ ¬ (𝐴 ∖ 𝐵) = 𝐴)) | |
5 | 3, 4 | mpbiran 991 | . . 3 ⊢ ((𝐴 ∖ 𝐵) ⊊ 𝐴 ↔ ¬ (𝐴 ∖ 𝐵) = 𝐴) |
6 | 5 | con2bii 346 | . 2 ⊢ ((𝐴 ∖ 𝐵) = 𝐴 ↔ ¬ (𝐴 ∖ 𝐵) ⊊ 𝐴) |
7 | 1, 2, 6 | 3bitri 286 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ¬ (𝐴 ∖ 𝐵) ⊊ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 = wceq 1632 ∖ cdif 3712 ∩ cin 3714 ⊆ wss 3715 ⊊ wpss 3716 ∅c0 4058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-v 3342 df-dif 3718 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 |
This theorem is referenced by: marypha1lem 8504 infeq5i 8706 wilthlem2 24994 topdifinffinlem 33506 |
Copyright terms: Public domain | W3C validator |