Theorem dfss5 3864

Description: Alternate definition of subclass relationship: a class A is a subclass of another class B iff each element of A is equal to an element of B. (Contributed by AV, 13-Nov-2020.)

Assertion

Ref	Expression
dfss5	|- ( A C_ B <-> A. x e. A E. y e. B x = y )

Distinct variable groups:   x, A   x, B, y

Allowed substitution hint:   A( y)

Proof of Theorem dfss5

Step	Hyp	Ref	Expression
1		dfss3 3592	. 2 |- ( A C_ B <-> A. x e. A x e. B )
2		clel5 3343	. . 3 |- ( x e. B <-> E. y e. B x = y )
3	2	ralbii 2980	. 2 |- ( A. x e. A x e. B <-> A. x e. A E. y e. B x = y )
4	1, 3	bitri 264	1 |- ( A C_ B <-> A. x e. A E. y e. B x = y )

Syntax hints:   <-> wb 196   e. wcel 1990   A.wral 2912   E.wrex 2913   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-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588

This theorem is referenced by:  usgrsscusgr  26356