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Mirrors > Home > ILE Home > Th. List > sseqin2 | GIF version |
Description: A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.) |
Ref | Expression |
---|---|
sseqin2 | ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss1 3285 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1332 ∩ cin 3075 ⊆ wss 3076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-in 3082 df-ss 3089 |
This theorem is referenced by: dfss4st 3314 resabs1 4856 rescnvcnv 5009 frecfnom 6306 fiintim 6825 nn0supp 9053 uzin 9382 iooval2 9728 fzval2 9824 dfphi2 11932 resttopon 12379 restabs 12383 restopnb 12389 txcnmpt 12481 |
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