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 | pm4.71 560 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))) | |
2 | eldif 3948 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
3 | 2 | bibi2i 340 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))) |
4 | 1, 3 | bitr4i 280 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))) |
5 | 4 | albii 1820 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))) |
6 | disj1 4403 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) | |
7 | dfcleq 2817 | . 2 ⊢ (𝐴 = (𝐴 ∖ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))) | |
8 | 5, 6, 7 | 3bitr4i 305 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 ∩ cin 3937 ∅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-ral 3145 df-v 3498 df-dif 3941 df-in 3945 df-nul 4294 |
This theorem is referenced by: disjel 4408 disj4 4410 uneqdifeq 4440 difprsn1 4735 diftpsn3 4737 ssunsn2 4762 orddif 6286 php 8703 hartogslem1 9008 infeq5i 9101 cantnfp1lem3 9145 dju1dif 9600 infdju1 9617 ssxr 10712 dprd2da 19166 dmdprdsplit2lem 19169 ablfac1eulem 19196 lbsextlem4 19935 opsrtoslem2 20267 alexsublem 22654 volun 24148 lhop1lem 24612 ex-dif 28204 difeq 30282 imadifxp 30353 disjdsct 30440 fzodif1 30518 carsgclctunlem1 31577 probun 31679 ballotlemfp1 31751 bj-disj2r 34342 topdifinfeq 34633 finixpnum 34879 lindsadd 34887 poimirlem11 34905 poimirlem12 34906 poimirlem13 34907 poimirlem14 34908 poimirlem16 34910 poimirlem18 34912 poimirlem21 34915 poimirlem22 34916 poimirlem27 34921 asindmre 34979 dffltz 39278 kelac2 39672 pwfi2f1o 39703 iccdifioo 41798 iccdifprioo 41799 |
Copyright terms: Public domain | W3C validator |