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Theorem bnj1279 28559
Description: Technical lemma for bnj60 28603. 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 3498 . . . . . . . . 9  |-  ( ( 
pred ( x ,  A ,  R )  i^i  E )  =/=  (/) 
<->  E. y  y  e.  (  pred ( x ,  A ,  R )  i^i  E ) )
2 elin 3392 . . . . . . . . . 10  |-  ( y  e.  (  pred (
x ,  A ,  R )  i^i  E
)  <->  ( y  e. 
pred ( x ,  A ,  R )  /\  y  e.  E
) )
32exbii 1573 . . . . . . . . 9  |-  ( E. y  y  e.  ( 
pred ( x ,  A ,  R )  i^i  E )  <->  E. y
( y  e.  pred ( x ,  A ,  R )  /\  y  e.  E ) )
41, 3bitri 240 . . . . . . . 8  |-  ( ( 
pred ( x ,  A ,  R )  i^i  E )  =/=  (/) 
<->  E. y ( y  e.  pred ( x ,  A ,  R )  /\  y  e.  E
) )
54biimpi 186 . . . . . . 7  |-  ( ( 
pred ( x ,  A ,  R )  i^i  E )  =/=  (/)  ->  E. y ( y  e.  pred ( x ,  A ,  R )  /\  y  e.  E
) )
6 df-bnj14 28225 . . . . . . . . 9  |-  pred (
x ,  A ,  R )  =  {
y  e.  A  | 
y R x }
76bnj1538 28398 . . . . . . . 8  |-  ( y  e.  pred ( x ,  A ,  R )  ->  y R x )
87anim1i 551 . . . . . . 7  |-  ( ( y  e.  pred (
x ,  A ,  R )  /\  y  e.  E )  ->  (
y R x  /\  y  e.  E )
)
95, 8bnj593 28285 . . . . . 6  |-  ( ( 
pred ( x ,  A ,  R )  i^i  E )  =/=  (/)  ->  E. y ( y R x  /\  y  e.  E ) )
1093ad2ant3 978 . . . . 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 ) )
11 nfv 1610 . . . . . . 7  |-  F/ y  x  e.  E
12 nfra1 2627 . . . . . . 7  |-  F/ y A. y  e.  E  -.  y R x
13 nfv 1610 . . . . . . 7  |-  F/ y (  pred ( x ,  A ,  R )  i^i  E )  =/=  (/)
1411, 12, 13nf3an 1803 . . . . . 6  |-  F/ y ( x  e.  E  /\  A. y  e.  E  -.  y R x  /\  (  pred ( x ,  A ,  R )  i^i  E )  =/=  (/) )
1514nfri 1766 . . . . 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
)  =/=  (/) ) )
1610, 15bnj1275 28357 . . . 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 ) )
17 simp2 956 . . . 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 )
18 simp12 986 . . . . 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 )
19 simp3 957 . . . . 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 )
2018, 19bnj1294 28361 . . . 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 )
2116, 17, 20bnj1304 28363 . . 3  |-  -.  (
x  e.  E  /\  A. y  e.  E  -.  y R x  /\  (  pred ( x ,  A ,  R )  i^i  E
)  =/=  (/) )
2221bnj1224 28345 . 2  |-  ( ( x  e.  E  /\  A. y  e.  E  -.  y R x )  ->  -.  (  pred ( x ,  A ,  R
)  i^i  E )  =/=  (/) )
23 nne 2483 . 2  |-  ( -.  (  pred ( x ,  A ,  R )  i^i  E )  =/=  (/) 
<->  (  pred ( x ,  A ,  R )  i^i  E )  =  (/) )
2422, 23sylib 188 1  |-  ( ( x  e.  E  /\  A. y  e.  E  -.  y R x )  -> 
(  pred ( x ,  A ,  R )  i^i  E )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1532    = wceq 1633    e. wcel 1701   {cab 2302    =/= wne 2479   A.wral 2577   E.wrex 2578   {crab 2581    i^i cin 3185    C_ wss 3186   (/)c0 3489   <.cop 3677   class class class wbr 4060   dom cdm 4726    |` cres 4728    Fn wfn 5287   ` cfv 5292    /\ w-bnj17 28222    predc-bnj14 28224    FrSe w-bnj15 28228
This theorem is referenced by:  bnj1311  28565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rab 2586  df-v 2824  df-dif 3189  df-in 3193  df-nul 3490  df-bnj14 28225
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