Theorem conss34OLD 38646

Description: Obsolete proof of complss 3751 as of 7-Aug-2021. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
conss34OLD	|- ( A C_ B <-> ( _V \ B ) C_ ( _V \ A ) )

Proof of Theorem conss34OLD

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		con34b 306	. . . 4 |- ( ( x e. A -> x e. B ) <-> ( -. x e. B -> -. x e. A ) )
2		compel 38641	. . . . 5 |- ( x e. ( _V \ B ) <-> -. x e. B )
3		compel 38641	. . . . 5 |- ( x e. ( _V \ A ) <-> -. x e. A )
4	2, 3	imbi12i 340	. . . 4 |- ( ( x e. ( _V \ B ) -> x e. ( _V \ A ) ) <-> ( -. x e. B -> -. x e. A ) )
5	1, 4	bitr4i 267	. . 3 |- ( ( x e. A -> x e. B ) <-> ( x e. ( _V \ B ) -> x e. ( _V \ A ) ) )
6	5	albii 1747	. 2 |- ( A. x ( x e. A -> x e. B ) <-> A. x ( x e. ( _V \ B ) -> x e. ( _V \ A ) ) )
7		dfss2 3591	. 2 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
8		dfss2 3591	. 2 |- ( ( _V \ B ) C_ ( _V \ A ) <-> A. x ( x e. ( _V \ B ) -> x e. ( _V \ A ) ) )
9	6, 7, 8	3bitr4i 292	1 |- ( A C_ B <-> ( _V \ B ) C_ ( _V \ A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   e. wcel 1990   _Vcvv 3200   \ cdif 3571   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-v 3202  df-dif 3577  df-in 3581  df-ss 3588

This theorem is referenced by: (None)