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Theorem onfrALTlem5 28606
Description: Lemma for onfrALT 28613. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem5  |-  ( [. ( a  i^i  x
)  /  b ]. ( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) 
<->  ( ( ( a  i^i  x )  C_  ( a  i^i  x
)  /\  ( a  i^i  x )  =/=  (/) )  ->  E. y  e.  (
a  i^i  x )
( ( a  i^i  x )  i^i  y
)  =  (/) ) )
Distinct variable groups:    a, b,
y    x, b, y

Proof of Theorem onfrALTlem5
StepHypRef Expression
1 vex 2804 . . . 4  |-  a  e. 
_V
21inex1 4171 . . 3  |-  ( a  i^i  x )  e. 
_V
3 sbcimg 3045 . . 3  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. ( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) 
<->  ( [. ( a  i^i  x )  / 
b ]. ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  [. ( a  i^i  x )  / 
b ]. E. y  e.  b  ( b  i^i  y )  =  (/) ) ) )
42, 3ax-mp 8 . 2  |-  ( [. ( a  i^i  x
)  /  b ]. ( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) 
<->  ( [. ( a  i^i  x )  / 
b ]. ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  [. ( a  i^i  x )  / 
b ]. E. y  e.  b  ( b  i^i  y )  =  (/) ) )
5 sbcang 3047 . . . . 5  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. ( b  C_  (
a  i^i  x )  /\  b  =/=  (/) )  <->  ( [. ( a  i^i  x
)  /  b ]. b  C_  ( a  i^i  x )  /\  [. (
a  i^i  x )  /  b ]. b  =/=  (/) ) ) )
62, 5ax-mp 8 . . . 4  |-  ( [. ( a  i^i  x
)  /  b ]. ( b  C_  (
a  i^i  x )  /\  b  =/=  (/) )  <->  ( [. ( a  i^i  x
)  /  b ]. b  C_  ( a  i^i  x )  /\  [. (
a  i^i  x )  /  b ]. b  =/=  (/) ) )
7 sseq1 3212 . . . . . . 7  |-  ( b  =  ( a  i^i  x )  ->  (
b  C_  ( a  i^i  x )  <->  ( a  i^i  x )  C_  (
a  i^i  x )
) )
87sbcieg 3036 . . . . . 6  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. b  C_  ( a  i^i  x )  <->  ( a  i^i  x )  C_  (
a  i^i  x )
) )
92, 8ax-mp 8 . . . . 5  |-  ( [. ( a  i^i  x
)  /  b ]. b  C_  ( a  i^i  x )  <->  ( a  i^i  x )  C_  (
a  i^i  x )
)
10 df-ne 2461 . . . . . . . 8  |-  ( b  =/=  (/)  <->  -.  b  =  (/) )
1110ax-gen 1536 . . . . . . 7  |-  A. b
( b  =/=  (/)  <->  -.  b  =  (/) )
12 sbcbi 28602 . . . . . . 7  |-  ( ( a  i^i  x )  e.  _V  ->  ( A. b ( b  =/=  (/) 
<->  -.  b  =  (/) )  ->  ( [. (
a  i^i  x )  /  b ]. b  =/=  (/)  <->  [. ( a  i^i  x )  /  b ].  -.  b  =  (/) ) ) )
132, 11, 12mp2 17 . . . . . 6  |-  ( [. ( a  i^i  x
)  /  b ]. b  =/=  (/)  <->  [. ( a  i^i  x )  /  b ].  -.  b  =  (/) )
14 sbcng 3044 . . . . . . . 8  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ].  -.  b  =  (/)  <->  -.  [. (
a  i^i  x )  /  b ]. b  =  (/) ) )
1514bicomd 192 . . . . . . 7  |-  ( ( a  i^i  x )  e.  _V  ->  ( -.  [. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  / 
b ].  -.  b  =  (/) ) )
162, 15ax-mp 8 . . . . . 6  |-  ( -. 
[. ( a  i^i  x )  /  b ]. b  =  (/)  <->  [. ( a  i^i  x )  / 
b ].  -.  b  =  (/) )
17 eqsbc3 3043 . . . . . . . 8  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =  (/) ) )
182, 17ax-mp 8 . . . . . . 7  |-  ( [. ( a  i^i  x
)  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =  (/) )
1918necon3bbii 2490 . . . . . 6  |-  ( -. 
[. ( a  i^i  x )  /  b ]. b  =  (/)  <->  ( a  i^i  x )  =/=  (/) )
2013, 16, 193bitr2i 264 . . . . 5  |-  ( [. ( a  i^i  x
)  /  b ]. b  =/=  (/)  <->  ( a  i^i  x )  =/=  (/) )
219, 20anbi12i 678 . . . 4  |-  ( (
[. ( a  i^i  x )  /  b ]. b  C_  ( a  i^i  x )  /\  [. ( a  i^i  x
)  /  b ]. b  =/=  (/) )  <->  ( (
a  i^i  x )  C_  ( a  i^i  x
)  /\  ( a  i^i  x )  =/=  (/) ) )
226, 21bitri 240 . . 3  |-  ( [. ( a  i^i  x
)  /  b ]. ( b  C_  (
a  i^i  x )  /\  b  =/=  (/) )  <->  ( (
a  i^i  x )  C_  ( a  i^i  x
)  /\  ( a  i^i  x )  =/=  (/) ) )
23 df-rex 2562 . . . . . 6  |-  ( E. y  e.  b  ( b  i^i  y )  =  (/)  <->  E. y ( y  e.  b  /\  (
b  i^i  y )  =  (/) ) )
2423ax-gen 1536 . . . . 5  |-  A. b
( E. y  e.  b  ( b  i^i  y )  =  (/)  <->  E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) )
25 sbcbi 28602 . . . . 5  |-  ( ( a  i^i  x )  e.  _V  ->  ( A. b ( E. y  e.  b  ( b  i^i  y )  =  (/)  <->  E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) )  ->  ( [. ( a  i^i  x
)  /  b ]. E. y  e.  b 
( b  i^i  y
)  =  (/)  <->  [. ( a  i^i  x )  / 
b ]. E. y ( y  e.  b  /\  ( b  i^i  y
)  =  (/) ) ) ) )
262, 24, 25mp2 17 . . . 4  |-  ( [. ( a  i^i  x
)  /  b ]. E. y  e.  b 
( b  i^i  y
)  =  (/)  <->  [. ( a  i^i  x )  / 
b ]. E. y ( y  e.  b  /\  ( b  i^i  y
)  =  (/) ) )
27 df-rex 2562 . . . . 5  |-  ( E. y  e.  ( a  i^i  x ) ( ( a  i^i  x
)  i^i  y )  =  (/)  <->  E. y ( y  e.  ( a  i^i  x )  /\  (
( a  i^i  x
)  i^i  y )  =  (/) ) )
28 sbcang 3047 . . . . . . . 8  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) 
<->  ( [. ( a  i^i  x )  / 
b ]. y  e.  b  /\  [. ( a  i^i  x )  / 
b ]. ( b  i^i  y )  =  (/) ) ) )
292, 28ax-mp 8 . . . . . . 7  |-  ( [. ( a  i^i  x
)  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) 
<->  ( [. ( a  i^i  x )  / 
b ]. y  e.  b  /\  [. ( a  i^i  x )  / 
b ]. ( b  i^i  y )  =  (/) ) )
30 sbcel2gv 3064 . . . . . . . . 9  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. y  e.  b  <->  y  e.  ( a  i^i  x
) ) )
312, 30ax-mp 8 . . . . . . . 8  |-  ( [. ( a  i^i  x
)  /  b ]. y  e.  b  <->  y  e.  ( a  i^i  x
) )
32 sbceqg 3110 . . . . . . . . . 10  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. ( b  i^i  y
)  =  (/)  <->  [_ ( a  i^i  x )  / 
b ]_ ( b  i^i  y )  =  [_ ( a  i^i  x
)  /  b ]_ (/) ) )
332, 32ax-mp 8 . . . . . . . . 9  |-  ( [. ( a  i^i  x
)  /  b ]. ( b  i^i  y
)  =  (/)  <->  [_ ( a  i^i  x )  / 
b ]_ ( b  i^i  y )  =  [_ ( a  i^i  x
)  /  b ]_ (/) )
34 csbing 3389 . . . . . . . . . . . 12  |-  ( ( a  i^i  x )  e.  _V  ->  [_ (
a  i^i  x )  /  b ]_ (
b  i^i  y )  =  ( [_ (
a  i^i  x )  /  b ]_ b  i^i  [_ ( a  i^i  x )  /  b ]_ y ) )
352, 34ax-mp 8 . . . . . . . . . . 11  |-  [_ (
a  i^i  x )  /  b ]_ (
b  i^i  y )  =  ( [_ (
a  i^i  x )  /  b ]_ b  i^i  [_ ( a  i^i  x )  /  b ]_ y )
36 csbvarg 3121 . . . . . . . . . . . . 13  |-  ( ( a  i^i  x )  e.  _V  ->  [_ (
a  i^i  x )  /  b ]_ b  =  ( a  i^i  x ) )
372, 36ax-mp 8 . . . . . . . . . . . 12  |-  [_ (
a  i^i  x )  /  b ]_ b  =  ( a  i^i  x )
38 csbconstg 3108 . . . . . . . . . . . . 13  |-  ( ( a  i^i  x )  e.  _V  ->  [_ (
a  i^i  x )  /  b ]_ y  =  y )
392, 38ax-mp 8 . . . . . . . . . . . 12  |-  [_ (
a  i^i  x )  /  b ]_ y  =  y
4037, 39ineq12i 3381 . . . . . . . . . . 11  |-  ( [_ ( a  i^i  x
)  /  b ]_ b  i^i  [_ ( a  i^i  x )  /  b ]_ y )  =  ( ( a  i^i  x
)  i^i  y )
4135, 40eqtri 2316 . . . . . . . . . 10  |-  [_ (
a  i^i  x )  /  b ]_ (
b  i^i  y )  =  ( ( a  i^i  x )  i^i  y )
42 csbconstg 3108 . . . . . . . . . . 11  |-  ( ( a  i^i  x )  e.  _V  ->  [_ (
a  i^i  x )  /  b ]_ (/)  =  (/) )
432, 42ax-mp 8 . . . . . . . . . 10  |-  [_ (
a  i^i  x )  /  b ]_ (/)  =  (/)
4441, 43eqeq12i 2309 . . . . . . . . 9  |-  ( [_ ( a  i^i  x
)  /  b ]_ ( b  i^i  y
)  =  [_ (
a  i^i  x )  /  b ]_ (/)  <->  ( (
a  i^i  x )  i^i  y )  =  (/) )
4533, 44bitri 240 . . . . . . . 8  |-  ( [. ( a  i^i  x
)  /  b ]. ( b  i^i  y
)  =  (/)  <->  ( (
a  i^i  x )  i^i  y )  =  (/) )
4631, 45anbi12i 678 . . . . . . 7  |-  ( (
[. ( a  i^i  x )  /  b ]. y  e.  b  /\  [. ( a  i^i  x )  /  b ]. ( b  i^i  y
)  =  (/) )  <->  ( y  e.  ( a  i^i  x
)  /\  ( (
a  i^i  x )  i^i  y )  =  (/) ) )
4729, 46bitri 240 . . . . . 6  |-  ( [. ( a  i^i  x
)  /  b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) ) 
<->  ( y  e.  ( a  i^i  x )  /\  ( ( a  i^i  x )  i^i  y )  =  (/) ) )
4847exbii 1572 . . . . 5  |-  ( E. y [. ( a  i^i  x )  / 
b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y ( y  e.  ( a  i^i  x )  /\  (
( a  i^i  x
)  i^i  y )  =  (/) ) )
49 sbcexg 3054 . . . . . . 7  |-  ( ( a  i^i  x )  e.  _V  ->  ( [. ( a  i^i  x
)  /  b ]. E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y [. (
a  i^i  x )  /  b ]. (
y  e.  b  /\  ( b  i^i  y
)  =  (/) ) ) )
5049bicomd 192 . . . . . 6  |-  ( ( a  i^i  x )  e.  _V  ->  ( E. y [. ( a  i^i  x )  / 
b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  [. ( a  i^i  x )  /  b ]. E. y ( y  e.  b  /\  (
b  i^i  y )  =  (/) ) ) )
512, 50ax-mp 8 . . . . 5  |-  ( E. y [. ( a  i^i  x )  / 
b ]. ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  [. ( a  i^i  x )  /  b ]. E. y ( y  e.  b  /\  (
b  i^i  y )  =  (/) ) )
5227, 48, 513bitr2ri 265 . . . 4  |-  ( [. ( a  i^i  x
)  /  b ]. E. y ( y  e.  b  /\  ( b  i^i  y )  =  (/) )  <->  E. y  e.  ( a  i^i  x ) ( ( a  i^i  x )  i^i  y
)  =  (/) )
5326, 52bitri 240 . . 3  |-  ( [. ( a  i^i  x
)  /  b ]. E. y  e.  b 
( b  i^i  y
)  =  (/)  <->  E. y  e.  ( a  i^i  x
) ( ( a  i^i  x )  i^i  y )  =  (/) )
5422, 53imbi12i 316 . 2  |-  ( (
[. ( a  i^i  x )  /  b ]. ( b  C_  (
a  i^i  x )  /\  b  =/=  (/) )  ->  [. ( a  i^i  x
)  /  b ]. E. y  e.  b 
( b  i^i  y
)  =  (/) )  <->  ( (
( a  i^i  x
)  C_  ( a  i^i  x )  /\  (
a  i^i  x )  =/=  (/) )  ->  E. y  e.  ( a  i^i  x
) ( ( a  i^i  x )  i^i  y )  =  (/) ) )
554, 54bitri 240 1  |-  ( [. ( a  i^i  x
)  /  b ]. ( ( b  C_  ( a  i^i  x
)  /\  b  =/=  (/) )  ->  E. y  e.  b  ( b  i^i  y )  =  (/) ) 
<->  ( ( ( a  i^i  x )  C_  ( a  i^i  x
)  /\  ( a  i^i  x )  =/=  (/) )  ->  E. y  e.  (
a  i^i  x )
( ( a  i^i  x )  i^i  y
)  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   _Vcvv 2801   [.wsbc 3004   [_csb 3094    i^i cin 3164    C_ wss 3165   (/)c0 3468
This theorem is referenced by:  onfrALTlem3  28608  onfrALTlem3VD  28979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-in 3172  df-ss 3179
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