Theorem ssn0rex 3936

Description: There is an element in a class with a nonempty subclass which is an element of the subclass. (Contributed by AV, 17-Dec-2020.)

Assertion

Ref	Expression
ssn0rex	|- ( ( A C_ B /\ A =/= (/) ) -> E. x e. B x e. A )

Distinct variable groups:   x, A   x, B

Proof of Theorem ssn0rex

Step	Hyp	Ref	Expression
1		ssrexv 3667	. 2 |- ( A C_ B -> ( E. x e. A x e. A -> E. x e. B x e. A ) )
2		n0rex 3935	. 2 |- ( A =/= (/) -> E. x e. A x e. A )
3	1, 2	impel 485	1 |- ( ( A C_ B /\ A =/= (/) ) -> E. x e. B x e. A )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   (/)c0 3915

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-nfc 2753  df-ne 2795  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  uhgrvd00  26430