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Mirrors > Home > ILE Home > Th. List > inss2 | GIF version |
Description: The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.) |
Ref | Expression |
---|---|
inss2 | ⊢ (A ∩ B) ⊆ B |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 3123 | . 2 ⊢ (B ∩ A) = (A ∩ B) | |
2 | inss1 3151 | . 2 ⊢ (B ∩ A) ⊆ B | |
3 | 1, 2 | eqsstr3i 2970 | 1 ⊢ (A ∩ B) ⊆ B |
Colors of variables: wff set class |
Syntax hints: ∩ cin 2910 ⊆ wss 2911 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-in 2918 df-ss 2925 |
This theorem is referenced by: difin0 3291 bnd2 3917 ordin 4088 relin2 4399 relres 4582 ssrnres 4706 cnvcnv 4716 funimaexg 4926 fnresin2 4957 ssimaex 5177 ffvresb 5271 ofrfval 5662 fnofval 5663 ofrval 5664 off 5666 ofres 5667 ofco 5671 offres 5704 tpostpos 5820 smores3 5849 tfrlem5 5871 tfrexlem 5889 erinxp 6116 ltrelpi 6308 peano5nni 7698 peano5set 9399 peano5setOLD 9400 |
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