Theorem sseqin2 3423

Description: A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)

Assertion

Ref	Expression
sseqin2	⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)

Proof of Theorem sseqin2

Step	Hyp	Ref	Expression
1		dfss1 3408	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)

Syntax hints:   ↔ wb 105   = wceq 1395   ∩ cin 3196   ⊆ wss 3197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210

This theorem is referenced by:  dfss4st  3437  resabs1  5034  mptimass  5081  rescnvcnv  5191  frecfnom  6553  fiintim  7101  nn0supp  9429  uzin  9763  iooval2  10119  fzval2  10215  suprzubdc  10464  bitsinv1  12481  dfphi2  12750  ressabsg  13117  resttopon  14853  restabs  14857  restopnb  14863  txcnmpt  14955