Theorem dfss1 3376

Description: A frequently-used variant of subclass definition df-ss 3178. (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 3178	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
2		incom 3364	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
3	2	eqeq1i 2212	. 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐴)
4	1, 3	bitri 184	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)

Syntax hints:   ↔ wb 105   = wceq 1372   ∩ cin 3164   ⊆ wss 3165

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178

This theorem is referenced by:  dfss5  3377  sseqin2  3391  onintexmid  4619  xpimasn  5128  fndmdif  5679  infiexmid  6956  ssfidc  7016  isumss  11621  znnen  12688  2omap  15796