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Mirrors > Home > MPE Home > Th. List > intiin | Structured version Visualization version GIF version |
Description: Class intersection in terms of indexed intersection. Definition in [Stoll] p. 44. (Contributed by NM, 28-Jun-1998.) |
Ref | Expression |
---|---|
intiin | ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfint2 4908 | . 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
2 | df-iin 4956 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝑥 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
3 | 1, 2 | eqtr4i 2769 | 1 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 {cab 2715 ∀wral 3063 ∩ cint 4906 ∩ ciin 4954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-ral 3064 df-int 4907 df-iin 4956 |
This theorem is referenced by: trint 5239 relint 5774 intpreima 7018 ixpint 8822 firest 17274 efger 19459 subdrgint 20223 rintopn 22210 intcld 22343 iundifdifd 31308 iundifdif 31309 |
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