Theorem elex2 2871

Description: If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)

Assertion

Ref	Expression
elex2	⊢ (A ∈ B → ∃x x ∈ B)

Distinct variable groups:   x,A   x,B

Proof of Theorem elex2

Step	Hyp	Ref	Expression
1		eleq1a 2422	. . 3 ⊢ (A ∈ B → (x = A → x ∈ B))
2	1	alrimiv 1631	. 2 ⊢ (A ∈ B → ∀x(x = A → x ∈ B))
3		elisset 2869	. 2 ⊢ (A ∈ B → ∃x x = A)
4		exim 1575	. 2 ⊢ (∀x(x = A → x ∈ B) → (∃x x = A → ∃x x ∈ B))
5	2, 3, 4	sylc 56	1 ⊢ (A ∈ B → ∃x x ∈ B)

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710

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-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861

This theorem is referenced by: (None)