Theorem ssrab3 3328

Description: Subclass relation for a restricted class abstraction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypothesis

Ref	Expression
ssrab3.1	⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}

Assertion

Ref	Expression
ssrab3	⊢ 𝐵 ⊆ 𝐴

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem ssrab3

Step	Hyp	Ref	Expression
1		ssrab3.1	. 2 ⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ 𝜑}
2		ssrab2 3327	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
3	1, 2	eqsstri 3274	1 ⊢ 𝐵 ⊆ 𝐴

Syntax hints:   = wceq 1398  {crab 2526   ⊆ wss 3214

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531  df-in 3220  df-ss 3227

This theorem is referenced by:  if0ss  3628  pcprecl  13012  pcprendvds  13013  4sqlem13m  13126  4sqlem14  13127  4sqlem17  13130  ballotfilemfmpn  13178  ballotfilemafi  13182  ballotfilembfi  13183  ballotfilemth  13225  nmzsubg  14011  nmznsg  14014  conjnmz  14080  conjnmzb  14081  nzrring  14413  lringnzr  14423  rrgeq0  14496  rrgss  14498  psrbagconf1o  14940  mpodvdsmulf1o  15970  fsumdvdsmul  15971  lgsfcl2  15991