Theorem cnvssOLD 5295

Description: Obsolete proof of cnvss 5294 as of 27-Apr-2021. Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
cnvssOLD	|- ( A C_ B -> `' A C_ `' B )

Proof of Theorem cnvssOLD

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3597	. . . 4 |- ( A C_ B -> ( <. y , x >. e. A -> <. y , x >. e. B ) )
2		df-br 4654	. . . 4 |- ( y A x <-> <. y , x >. e. A )
3		df-br 4654	. . . 4 |- ( y B x <-> <. y , x >. e. B )
4	1, 2, 3	3imtr4g 285	. . 3 |- ( A C_ B -> ( y A x -> y B x ) )
5	4	ssopab2dv 5004	. 2 |- ( A C_ B -> { <. x , y >. | y A x } C_ { <. x , y >. | y B x } )
6		df-cnv 5122	. 2 |- `' A = { <. x , y >. | y A x }
7		df-cnv 5122	. 2 |- `' B = { <. x , y >. | y B x }
8	5, 6, 7	3sstr4g 3646	1 |- ( A C_ B -> `' A C_ `' B )

Syntax hints:   -> wi 4   e. wcel 1990   C_ wss 3574   <.cop 4183   class class class wbr 4653   {copab 4712   `'ccnv 5113

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-in 3581  df-ss 3588  df-br 4654  df-opab 4713  df-cnv 5122

This theorem is referenced by: (None)