| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > dfin5 | Structured version Visualization version GIF version | ||
| Description: Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.) |
| Ref | Expression |
|---|---|
| dfin5 | ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-in 3958 | . 2 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
| 2 | df-rab 3437 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
| 3 | 1, 2 | eqtr4i 2768 | 1 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 {crab 3436 ∩ cin 3950 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-rab 3437 df-in 3958 |
| This theorem is referenced by: incom 4209 ineq1 4213 nfinOLD 4225 rabbi2dva 4226 dfss7 4251 dfepfr 5669 epfrc 5670 pmtrmvd 19474 ablfaclem3 20107 mretopd 23100 ptclsg 23623 xkopt 23663 iscmet3 25327 xrlimcnp 27011 ppiub 27248 xppreima 32655 fpwrelmapffs 32745 orvcelval 34471 sstotbnd2 37781 glbconN 39378 glbconNOLD 39379 2polssN 39917 rfovcnvf1od 44017 fsovcnvlem 44026 ntrneifv3 44095 ntrneifv4 44098 clsneifv3 44123 clsneifv4 44124 neicvgfv 44134 inpw 48738 |
| Copyright terms: Public domain | W3C validator |