Theorem dfss1 3461

Description: A frequently-used variant of subclass definition df-ss 3260. (Contributed by NM, 10-Jan-2015.)

Assertion

Ref	Expression
dfss1	⊢ (A ⊆ B ↔ (B ∩ A) = A)

Proof of Theorem dfss1

Step	Hyp	Ref	Expression
1		df-ss 3260	. 2 ⊢ (A ⊆ B ↔ (A ∩ B) = A)
2		incom 3449	. . 3 ⊢ (A ∩ B) = (B ∩ A)
3	2	eqeq1i 2360	. 2 ⊢ ((A ∩ B) = A ↔ (B ∩ A) = A)
4	1, 3	bitri 240	1 ⊢ (A ⊆ B ↔ (B ∩ A) = A)

Syntax hints:   ↔ wb 176   = wceq 1642   ∩ cin 3209   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  dfss5  3462  sseqin2  3475