Theorem sseliALT 4791

Description: Alternate proof of sseli 3599 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3600. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sseliALT.1	|- A C_ B

Assertion

Ref	Expression
sseliALT	|- ( C e. A -> C e. B )

Proof of Theorem sseliALT

Step	Hyp	Ref	Expression
1		biidd 252	. 2 |- ( A = if ( C e. A , A , { (/) } ) -> ( C e. B <-> C e. B ) )
2		eleq2 2690	. 2 |- ( B = if ( C e. A , B , { (/) } ) -> ( C e. B <-> C e. if ( C e. A , B , { (/) } ) ) )
3		eleq1 2689	. 2 |- ( C = if ( C e. A , C , (/) ) -> ( C e. if ( C e. A , B , { (/) } ) <-> if ( C e. A , C , (/) ) e. if ( C e. A , B , { (/) } ) ) )
4		sseq1 3626	. . . 4 |- ( A = if ( C e. A , A , { (/) } ) -> ( A C_ B <-> if ( C e. A , A , { (/) } ) C_ B ) )
5		sseq2 3627	. . . 4 |- ( B = if ( C e. A , B , { (/) } ) -> ( if ( C e. A , A , { (/) } ) C_ B <-> if ( C e. A , A , { (/) } ) C_ if ( C e. A , B , { (/) } ) ) )
6		biidd 252	. . . 4 |- ( C = if ( C e. A , C , (/) ) -> ( if ( C e. A , A , { (/) } ) C_ if ( C e. A , B , { (/) } ) <-> if ( C e. A , A , { (/) } ) C_ if ( C e. A , B , { (/) } ) ) )
7		sseq1 3626	. . . 4 |- ( { (/) } = if ( C e. A , A , { (/) } ) -> ( { (/) } C_ { (/) } <-> if ( C e. A , A , { (/) } ) C_ { (/) } ) )
8		sseq2 3627	. . . 4 |- ( { (/) } = if ( C e. A , B , { (/) } ) -> ( if ( C e. A , A , { (/) } ) C_ { (/) } <-> if ( C e. A , A , { (/) } ) C_ if ( C e. A , B , { (/) } ) ) )
9		biidd 252	. . . 4 |- ( (/) = if ( C e. A , C , (/) ) -> ( if ( C e. A , A , { (/) } ) C_ if ( C e. A , B , { (/) } ) <-> if ( C e. A , A , { (/) } ) C_ if ( C e. A , B , { (/) } ) ) )
10		sseliALT.1	. . . 4 |- A C_ B
11		ssid 3624	. . . 4 |- { (/) } C_ { (/) }
12	4, 5, 6, 7, 8, 9, 10, 11	keephyp3v 4154	. . 3 |- if ( C e. A , A , { (/) } ) C_ if ( C e. A , B , { (/) } )
13		eleq2 2690	. . . 4 |- ( A = if ( C e. A , A , { (/) } ) -> ( C e. A <-> C e. if ( C e. A , A , { (/) } ) ) )
14		biidd 252	. . . 4 |- ( B = if ( C e. A , B , { (/) } ) -> ( C e. if ( C e. A , A , { (/) } ) <-> C e. if ( C e. A , A , { (/) } ) ) )
15		eleq1 2689	. . . 4 |- ( C = if ( C e. A , C , (/) ) -> ( C e. if ( C e. A , A , { (/) } ) <-> if ( C e. A , C , (/) ) e. if ( C e. A , A , { (/) } ) ) )
16		eleq2 2690	. . . 4 |- ( { (/) } = if ( C e. A , A , { (/) } ) -> ( (/) e. { (/) } <-> (/) e. if ( C e. A , A , { (/) } ) ) )
17		biidd 252	. . . 4 |- ( { (/) } = if ( C e. A , B , { (/) } ) -> ( (/) e. if ( C e. A , A , { (/) } ) <-> (/) e. if ( C e. A , A , { (/) } ) ) )
18		eleq1 2689	. . . 4 |- ( (/) = if ( C e. A , C , (/) ) -> ( (/) e. if ( C e. A , A , { (/) } ) <-> if ( C e. A , C , (/) ) e. if ( C e. A , A , { (/) } ) ) )
19		0ex 4790	. . . . 5 |- (/) e. _V
20	19	snid 4208	. . . 4 |- (/) e. { (/) }
21	13, 14, 15, 16, 17, 18, 20	elimhyp3v 4148	. . 3 |- if ( C e. A , C , (/) ) e. if ( C e. A , A , { (/) } )
22	12, 21	sselii 3600	. 2 |- if ( C e. A , C , (/) ) e. if ( C e. A , B , { (/) } )
23	1, 2, 3, 22	dedth3v 4144	1 |- ( C e. A -> C e. B )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177

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  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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  df-nul 3916  df-if 4087  df-sn 4178

This theorem is referenced by: (None)