Theorem dfss5 3332

Description: Another definition of subclasshood. Similar to df-ss 3134, dfss 3135, and dfss1 3331. (Contributed by David Moews, 1-May-2017.)

Assertion

Ref	Expression
dfss5	⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐵 ∩ 𝐴))

Proof of Theorem dfss5

Step	Hyp	Ref	Expression
1		dfss1 3331	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
2		eqcom 2172	. 2 ⊢ ((𝐵 ∩ 𝐴) = 𝐴 ↔ 𝐴 = (𝐵 ∩ 𝐴))
3	1, 2	bitri 183	1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐵 ∩ 𝐴))

Syntax hints:   ↔ wb 104   = wceq 1348   ∩ cin 3120   ⊆ wss 3121

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134

This theorem is referenced by:  nninfdcex  11908  nnmindc  11989  nnminle  11990