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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 3981 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)) | |
2 | 1 | ss2abdv 3991 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} ⊆ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}) |
3 | dfint2 4840 | . 2 ⊢ ∩ 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} | |
4 | dfint2 4840 | . 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
5 | 2, 3, 4 | 3sstr4g 3960 | 1 ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 {cab 2776 ∀wral 3106 ⊆ wss 3881 ∩ cint 4838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-v 3443 df-in 3888 df-ss 3898 df-int 4839 |
This theorem is referenced by: uniintsn 4875 intabs 5209 fiss 8872 tc2 9168 tcss 9170 tcel 9171 rankval4 9280 cfub 9660 cflm 9661 cflecard 9664 fin23lem26 9736 clsslem 14335 mrcss 16879 lspss 19749 lbsextlem3 19925 aspss 20563 clsss 21659 1stcfb 22050 ufinffr 22534 spanss 29131 ss2mcls 32928 pclssN 37190 dochspss 38674 clss2lem 40311 |
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