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Theorem sseliALT 4571
Description: Alternate proof of sseli 3493 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3494. (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
StepHypRef Expression
1 biidd 237 . 2  |-  ( A  =  if ( C  e.  A ,  A ,  { (/) } )  -> 
( C  e.  B  <->  C  e.  B ) )
2 eleq2 2533 . 2  |-  ( B  =  if ( C  e.  A ,  B ,  { (/) } )  -> 
( C  e.  B  <->  C  e.  if ( C  e.  A ,  B ,  { (/) } ) ) )
3 eleq1 2532 . 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 3518 . . . 4  |-  ( A  =  if ( C  e.  A ,  A ,  { (/) } )  -> 
( A  C_  B  <->  if ( C  e.  A ,  A ,  { (/) } )  C_  B )
)
5 sseq2 3519 . . . 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 237 . . . 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 3518 . . . 4  |-  ( {
(/) }  =  if ( C  e.  A ,  A ,  { (/) } )  ->  ( { (/)
}  C_  { (/) }  <->  if ( C  e.  A ,  A ,  { (/) } ) 
C_  { (/) } ) )
8 sseq2 3519 . . . 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 237 . . . 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 3516 . . . 4  |-  { (/) } 
C_  { (/) }
124, 5, 6, 7, 8, 9, 10, 11keephyp3v 3999 . . 3  |-  if ( C  e.  A ,  A ,  { (/) } ) 
C_  if ( C  e.  A ,  B ,  { (/) } )
13 eleq2 2533 . . . 4  |-  ( A  =  if ( C  e.  A ,  A ,  { (/) } )  -> 
( C  e.  A  <->  C  e.  if ( C  e.  A ,  A ,  { (/) } ) ) )
14 biidd 237 . . . 4  |-  ( B  =  if ( C  e.  A ,  B ,  { (/) } )  -> 
( C  e.  if ( C  e.  A ,  A ,  { (/) } )  <->  C  e.  if ( C  e.  A ,  A ,  { (/) } ) ) )
15 eleq1 2532 . . . 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 2533 . . . 4  |-  ( {
(/) }  =  if ( C  e.  A ,  A ,  { (/) } )  ->  ( (/)  e.  { (/)
}  <->  (/)  e.  if ( C  e.  A ,  A ,  { (/) } ) ) )
17 biidd 237 . . . 4  |-  ( {
(/) }  =  if ( C  e.  A ,  B ,  { (/) } )  ->  ( (/)  e.  if ( C  e.  A ,  A ,  { (/) } )  <->  (/)  e.  if ( C  e.  A ,  A ,  { (/) } ) ) )
18 eleq1 2532 . . . 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 4570 . . . . 5  |-  (/)  e.  _V
2019snid 4048 . . . 4  |-  (/)  e.  { (/)
}
2113, 14, 15, 16, 17, 18, 20elimhyp3v 3993 . . 3  |-  if ( C  e.  A ,  C ,  (/) )  e.  if ( C  e.  A ,  A ,  { (/) } )
2212, 21sselii 3494 . 2  |-  if ( C  e.  A ,  C ,  (/) )  e.  if ( C  e.  A ,  B ,  { (/) } )
231, 2, 3, 22dedth3v 3989 1  |-  ( C  e.  A  ->  C  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762    C_ wss 3469   (/)c0 3778   ifcif 3932   {csn 4020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-nul 4569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-v 3108  df-dif 3472  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021
This theorem is referenced by: (None)
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