![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > disj3 | Structured version Visualization version GIF version |
Description: Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.) |
Ref | Expression |
---|---|
disj3 | ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.71 561 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))) | |
2 | eldif 3891 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
3 | 2 | bibi2i 341 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))) |
4 | 1, 3 | bitr4i 281 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))) |
5 | 4 | albii 1821 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))) |
6 | disj1 4358 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) | |
7 | dfcleq 2792 | . 2 ⊢ (𝐴 = (𝐴 ∖ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))) | |
8 | 5, 6, 7 | 3bitr4i 306 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ∩ cin 3880 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-v 3443 df-dif 3884 df-in 3888 df-nul 4244 |
This theorem is referenced by: disjel 4364 disj4 4366 uneqdifeq 4396 difprsn1 4693 diftpsn3 4695 ssunsn2 4720 orddif 6252 php 8685 hartogslem1 8990 infeq5i 9083 cantnfp1lem3 9127 dju1dif 9583 infdju1 9600 ssxr 10699 dprd2da 19157 dmdprdsplit2lem 19160 ablfac1eulem 19187 lbsextlem4 19926 opsrtoslem2 20724 alexsublem 22649 volun 24149 lhop1lem 24616 ex-dif 28208 difeq 30289 imadifxp 30364 disjdsct 30462 fzodif1 30542 carsgclctunlem1 31685 probun 31787 ballotlemfp1 31859 bj-disj2r 34464 topdifinfeq 34767 finixpnum 35042 lindsadd 35050 poimirlem11 35068 poimirlem12 35069 poimirlem13 35070 poimirlem14 35071 poimirlem16 35073 poimirlem18 35075 poimirlem21 35078 poimirlem22 35079 poimirlem27 35084 asindmre 35140 dffltz 39615 kelac2 40009 pwfi2f1o 40040 iccdifioo 42152 iccdifprioo 42153 |
Copyright terms: Public domain | W3C validator |