Theorem bnj1361 30899

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1361.1	|- ( ph -> A. x ( x e. A -> x e. B ) )

Assertion

Ref	Expression
bnj1361	|- ( ph -> A C_ B )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem bnj1361

Step	Hyp	Ref	Expression
1		bnj1361.1	. 2 |- ( ph -> A. x ( x e. A -> x e. B ) )
2		dfss2 3591	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
3	1, 2	sylibr 224	1 |- ( ph -> A C_ B )

Syntax hints:   -> wi 4   A.wal 1481   e. wcel 1990   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588

This theorem is referenced by: (None)