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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 4033 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)) | |
2 | 1 | ss2abdv 4044 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} ⊆ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}) |
3 | dfint2 4878 | . 2 ⊢ ∩ 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} | |
4 | dfint2 4878 | . 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
5 | 2, 3, 4 | 3sstr4g 4012 | 1 ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 {cab 2799 ∀wral 3138 ⊆ wss 3936 ∩ cint 4876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-in 3943 df-ss 3952 df-int 4877 |
This theorem is referenced by: uniintsn 4913 intabs 5245 fiss 8888 tc2 9184 tcss 9186 tcel 9187 rankval4 9296 cfub 9671 cflm 9672 cflecard 9675 fin23lem26 9747 clsslem 14344 mrcss 16887 lspss 19756 lbsextlem3 19932 aspss 20106 clsss 21662 1stcfb 22053 ufinffr 22537 spanss 29125 ss2mcls 32815 pclssN 37045 dochspss 38529 clss2lem 39991 |
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