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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 4869 | . 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)} | |
2 | df-ral 3143 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)) | |
3 | 2 | abbii 2886 | . 2 ⊢ {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)} |
4 | 1, 3 | eqtr4i 2847 | 1 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 = wceq 1533 ∈ wcel 2110 {cab 2799 ∀wral 3138 ∩ cint 4868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-9 2120 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-sb 2066 df-clab 2800 df-cleq 2814 df-ral 3143 df-int 4869 |
This theorem is referenced by: inteq 4871 elintg 4876 nfint 4878 intss 4889 intiin 4975 dfint3 33408 |
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