Theorem trssOLD 4762

Description: Obsolete proof of trss 4761 as of 26-Jul-2021. (Contributed by NM, 7-Aug-1994.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
trssOLD	|- ( Tr A -> ( B e. A -> B C_ A ) )

Proof of Theorem trssOLD

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
2		sseq1 3626	. . . . 5 |- ( x = B -> ( x C_ A <-> B C_ A ) )
3	1, 2	imbi12d 334	. . . 4 |- ( x = B -> ( ( x e. A -> x C_ A ) <-> ( B e. A -> B C_ A ) ) )
4	3	imbi2d 330	. . 3 |- ( x = B -> ( ( Tr A -> ( x e. A -> x C_ A ) ) <-> ( Tr A -> ( B e. A -> B C_ A ) ) ) )
5		dftr3 4756	. . . 4 |- ( Tr A <-> A. x e. A x C_ A )
6		rsp 2929	. . . 4 |- ( A. x e. A x C_ A -> ( x e. A -> x C_ A ) )
7	5, 6	sylbi 207	. . 3 |- ( Tr A -> ( x e. A -> x C_ A ) )
8	4, 7	vtoclg 3266	. 2 |- ( B e. A -> ( Tr A -> ( B e. A -> B C_ A ) ) )
9	8	pm2.43b 55	1 |- ( Tr A -> ( B e. A -> B C_ A ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   Tr wtr 4752

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-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-tr 4753

This theorem is referenced by: (None)