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Theorem bnj1279 33554
Description: Technical lemma for bnj60 33598. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1279.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1279.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1279.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1279.4  |-  D  =  ( dom  g  i^i 
dom  h )
bnj1279.5  |-  E  =  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
bnj1279.6  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
bnj1279.7  |-  ( ps  <->  (
ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )
Assertion
Ref Expression
bnj1279  |-  ( ( x  e.  E  /\  A. y  e.  E  -.  y R x )  -> 
(  pred ( x ,  A ,  R )  i^i  E )  =  (/) )
Distinct variable groups:    y, A    y, E    y, R    x, y
Allowed substitution hints:    ph( x, y, f, g, h, d)    ps( x, y, f, g, h, d)    A( x, f, g, h, d)    B( x, y, f, g, h, d)    C( x, y, f, g, h, d)    D( x, y, f, g, h, d)    R( x, f, g, h, d)    E( x, f, g, h, d)    G( x, y, f, g, h, d)    Y( x, y, f, g, h, d)

Proof of Theorem bnj1279
StepHypRef Expression
1 n0 3799 . . . . . . . 8  |-  ( ( 
pred ( x ,  A ,  R )  i^i  E )  =/=  (/) 
<->  E. y  y  e.  (  pred ( x ,  A ,  R )  i^i  E ) )
2 elin 3692 . . . . . . . . 9  |-  ( y  e.  (  pred (
x ,  A ,  R )  i^i  E
)  <->  ( y  e. 
pred ( x ,  A ,  R )  /\  y  e.  E
) )
32exbii 1644 . . . . . . . 8  |-  ( E. y  y  e.  ( 
pred ( x ,  A ,  R )  i^i  E )  <->  E. y
( y  e.  pred ( x ,  A ,  R )  /\  y  e.  E ) )
41, 3sylbb 197 . . . . . . 7  |-  ( ( 
pred ( x ,  A ,  R )  i^i  E )  =/=  (/)  ->  E. y ( y  e.  pred ( x ,  A ,  R )  /\  y  e.  E
) )
5 df-bnj14 33222 . . . . . . . . 9  |-  pred (
x ,  A ,  R )  =  {
y  e.  A  | 
y R x }
65bnj1538 33393 . . . . . . . 8  |-  ( y  e.  pred ( x ,  A ,  R )  ->  y R x )
76anim1i 568 . . . . . . 7  |-  ( ( y  e.  pred (
x ,  A ,  R )  /\  y  e.  E )  ->  (
y R x  /\  y  e.  E )
)
84, 7bnj593 33282 . . . . . 6  |-  ( ( 
pred ( x ,  A ,  R )  i^i  E )  =/=  (/)  ->  E. y ( y R x  /\  y  e.  E ) )
983ad2ant3 1019 . . . . 5  |-  ( ( x  e.  E  /\  A. y  e.  E  -.  y R x  /\  (  pred ( x ,  A ,  R )  i^i  E
)  =/=  (/) )  ->  E. y ( y R x  /\  y  e.  E ) )
10 nfv 1683 . . . . . . 7  |-  F/ y  x  e.  E
11 nfra1 2848 . . . . . . 7  |-  F/ y A. y  e.  E  -.  y R x
12 nfv 1683 . . . . . . 7  |-  F/ y (  pred ( x ,  A ,  R )  i^i  E )  =/=  (/)
1310, 11, 12nf3an 1877 . . . . . 6  |-  F/ y ( x  e.  E  /\  A. y  e.  E  -.  y R x  /\  (  pred ( x ,  A ,  R )  i^i  E )  =/=  (/) )
1413nfri 1822 . . . . 5  |-  ( ( x  e.  E  /\  A. y  e.  E  -.  y R x  /\  (  pred ( x ,  A ,  R )  i^i  E
)  =/=  (/) )  ->  A. y ( x  e.  E  /\  A. y  e.  E  -.  y R x  /\  (  pred ( x ,  A ,  R )  i^i  E
)  =/=  (/) ) )
159, 14bnj1275 33352 . . . 4  |-  ( ( x  e.  E  /\  A. y  e.  E  -.  y R x  /\  (  pred ( x ,  A ,  R )  i^i  E
)  =/=  (/) )  ->  E. y ( ( x  e.  E  /\  A. y  e.  E  -.  y R x  /\  (  pred ( x ,  A ,  R )  i^i  E
)  =/=  (/) )  /\  y R x  /\  y  e.  E ) )
16 simp2 997 . . . 4  |-  ( ( ( x  e.  E  /\  A. y  e.  E  -.  y R x  /\  (  pred ( x ,  A ,  R )  i^i  E )  =/=  (/) )  /\  y R x  /\  y  e.  E )  ->  y R x )
17 simp12 1027 . . . . 5  |-  ( ( ( x  e.  E  /\  A. y  e.  E  -.  y R x  /\  (  pred ( x ,  A ,  R )  i^i  E )  =/=  (/) )  /\  y R x  /\  y  e.  E )  ->  A. y  e.  E  -.  y R x )
18 simp3 998 . . . . 5  |-  ( ( ( x  e.  E  /\  A. y  e.  E  -.  y R x  /\  (  pred ( x ,  A ,  R )  i^i  E )  =/=  (/) )  /\  y R x  /\  y  e.  E )  ->  y  e.  E )
1917, 18bnj1294 33356 . . . 4  |-  ( ( ( x  e.  E  /\  A. y  e.  E  -.  y R x  /\  (  pred ( x ,  A ,  R )  i^i  E )  =/=  (/) )  /\  y R x  /\  y  e.  E )  ->  -.  y R x )
2015, 16, 19bnj1304 33358 . . 3  |-  -.  (
x  e.  E  /\  A. y  e.  E  -.  y R x  /\  (  pred ( x ,  A ,  R )  i^i  E
)  =/=  (/) )
2120bnj1224 33340 . 2  |-  ( ( x  e.  E  /\  A. y  e.  E  -.  y R x )  ->  -.  (  pred ( x ,  A ,  R
)  i^i  E )  =/=  (/) )
22 nne 2668 . 2  |-  ( -.  (  pred ( x ,  A ,  R )  i^i  E )  =/=  (/) 
<->  (  pred ( x ,  A ,  R )  i^i  E )  =  (/) )
2321, 22sylib 196 1  |-  ( ( x  e.  E  /\  A. y  e.  E  -.  y R x )  -> 
(  pred ( x ,  A ,  R )  i^i  E )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2817   E.wrex 2818   {crab 2821    i^i cin 3480    C_ wss 3481   (/)c0 3790   <.cop 4039   class class class wbr 4453   dom cdm 5005    |` cres 5007    Fn wfn 5589   ` cfv 5594    /\ w-bnj17 33219    predc-bnj14 33221    FrSe w-bnj15 33225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rab 2826  df-v 3120  df-dif 3484  df-in 3488  df-nul 3791  df-bnj14 33222
This theorem is referenced by:  bnj1311  33560
  Copyright terms: Public domain W3C validator