Theorem dfss5 3338

Description: Another definition of subclasshood. Similar to df-ss 3140, dfss 3141, and dfss1 3337. (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 3337	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
2		eqcom 2177	. 2 ⊢ ((𝐵 ∩ 𝐴) = 𝐴 ↔ 𝐴 = (𝐵 ∩ 𝐴))
3	1, 2	bitri 184	1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐵 ∩ 𝐴))

Syntax hints:   ↔ wb 105   = wceq 1353   ∩ cin 3126   ⊆ wss 3127

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-in 3133  df-ss 3140

This theorem is referenced by:  nninfdcex  11921  nnmindc  12002  nnminle  12003