Theorem n0rex 3935

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

Assertion

Ref	Expression
n0rex	|- ( A =/= (/) -> E. x e. A x e. A )

Distinct variable group:   x, A

Proof of Theorem n0rex

Step	Hyp	Ref	Expression
1		id 22	. . . 4 |- ( x e. A -> x e. A )
2	1	ancli 574	. . 3 |- ( x e. A -> ( x e. A /\ x e. A ) )
3	2	eximi 1762	. 2 |- ( E. x x e. A -> E. x ( x e. A /\ x e. A ) )
4		n0 3931	. 2 |- ( A =/= (/) <-> E. x x e. A )
5		df-rex 2918	. 2 |- ( E. x e. A x e. A <-> E. x ( x e. A /\ x e. A ) )
6	3, 4, 5	3imtr4i 281	1 |- ( A =/= (/) -> E. x e. A x e. A )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   (/)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-nul 3916

This theorem is referenced by:  ssn0rex  3936