Theorem alscn0d 42541

Description: Deduction rule: Given "all some" applied to a class, the class is not the empty set. (Contributed by David A. Wheeler, 23-Oct-2018.)

Hypothesis

Ref	Expression
alscn0d.1	|- ( ph -> A.! x e. A ps )

Assertion

Ref	Expression
alscn0d	|- ( ph -> A =/= (/) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem alscn0d

Step	Hyp	Ref	Expression
1		alscn0d.1	. . 3 |- ( ph -> A.! x e. A ps )
2	1	alsc2d 42540	. 2 |- ( ph -> E. x x e. A )
3		n0 3931	. 2 |- ( A =/= (/) <-> E. x x e. A )
4	2, 3	sylibr 224	1 |- ( ph -> A =/= (/) )

Syntax hints:   -> wi 4   E.wex 1704   e. wcel 1990   =/= wne 2794   (/)c0 3915   A.!walsc 42533

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-v 3202  df-dif 3577  df-nul 3916  df-alsc 42535

This theorem is referenced by: (None)