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Mirrors > Home > MPE Home > Th. List > intss | Structured version Visualization version GIF version |
Description: Intersection of subclasses. (Contributed by NM, 14-Oct-1999.) (Proof shortened by OpenAI, 25-Mar-2020.) |
Ref | Expression |
---|---|
intss | ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssralv 4064 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)) | |
2 | 1 | ss2abdv 4076 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} ⊆ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}) |
3 | dfint2 4953 | . 2 ⊢ ∩ 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} | |
4 | dfint2 4953 | . 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
5 | 2, 3, 4 | 3sstr4g 4041 | 1 ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 {cab 2712 ∀wral 3059 ⊆ wss 3963 ∩ cint 4951 |
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 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-ral 3060 df-ss 3980 df-int 4952 |
This theorem is referenced by: uniintsn 4990 intabs 5355 cofon1 8709 naddssim 8722 fiss 9462 tc2 9780 tcss 9782 tcel 9783 rankval4 9905 cfub 10287 cflm 10288 cflecard 10291 fin23lem26 10363 clsslem 15020 mrcss 17661 lspss 21000 lbsextlem3 21180 aspss 21915 clsss 23078 1stcfb 23469 ufinffr 23953 cofcut1 27969 spanss 31377 fldgenss 33298 ss2mcls 35553 pclssN 39877 dochspss 41361 clss2lem 43601 |
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