Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dfint2 | Structured version Visualization version GIF version |
Description: Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.) |
Ref | Expression |
---|---|
dfint2 | ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-int 4411 | . 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)} | |
2 | df-ral 2901 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)) | |
3 | 2 | abbii 2726 | . 2 ⊢ {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)} |
4 | 1, 3 | eqtr4i 2635 | 1 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 = wceq 1475 ∈ wcel 1977 {cab 2596 ∀wral 2896 ∩ cint 4410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-ral 2901 df-int 4411 |
This theorem is referenced by: inteq 4413 elintg 4418 nfint 4421 intss 4433 intiin 4510 dfint3 31229 |
Copyright terms: Public domain | W3C validator |