Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1143 Unicode version

Theorem bnj1143 29138
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1143  |-  U_ x  e.  A  B  C_  B
Distinct variable groups:    x, A    x, B

Proof of Theorem bnj1143
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iun 3923 . . . 4  |-  U_ x  e.  A  B  =  { y  |  E. x  e.  A  y  e.  B }
2 notnot 282 . . . . . . . 8  |-  ( A  =  (/)  <->  -.  -.  A  =  (/) )
3 neq0 3478 . . . . . . . 8  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
42, 3xchbinx 301 . . . . . . 7  |-  ( A  =  (/)  <->  -.  E. x  x  e.  A )
5 df-rex 2562 . . . . . . . . 9  |-  ( E. x  e.  A  z  e.  B  <->  E. x
( x  e.  A  /\  z  e.  B
) )
6 exsimpl 1582 . . . . . . . . 9  |-  ( E. x ( x  e.  A  /\  z  e.  B )  ->  E. x  x  e.  A )
75, 6sylbi 187 . . . . . . . 8  |-  ( E. x  e.  A  z  e.  B  ->  E. x  x  e.  A )
87con3i 127 . . . . . . 7  |-  ( -. 
E. x  x  e.  A  ->  -.  E. x  e.  A  z  e.  B )
94, 8sylbi 187 . . . . . 6  |-  ( A  =  (/)  ->  -.  E. x  e.  A  z  e.  B )
109alrimiv 1621 . . . . 5  |-  ( A  =  (/)  ->  A. z  -.  E. x  e.  A  z  e.  B )
11 notnot 282 . . . . . . 7  |-  ( { y  |  E. x  e.  A  y  e.  B }  =  (/)  <->  -.  -.  {
y  |  E. x  e.  A  y  e.  B }  =  (/) )
12 neq0 3478 . . . . . . . 8  |-  ( -. 
U_ x  e.  A  B  =  (/)  <->  E. z 
z  e.  U_ x  e.  A  B )
131eqeq1i 2303 . . . . . . . . 9  |-  ( U_ x  e.  A  B  =  (/)  <->  { y  |  E. x  e.  A  y  e.  B }  =  (/) )
1413notbii 287 . . . . . . . 8  |-  ( -. 
U_ x  e.  A  B  =  (/)  <->  -.  { y  |  E. x  e.  A  y  e.  B }  =  (/) )
15 df-iun 3923 . . . . . . . . . 10  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
1615eleq2i 2360 . . . . . . . . 9  |-  ( z  e.  U_ x  e.  A  B  <->  z  e.  { z  |  E. x  e.  A  z  e.  B } )
1716exbii 1572 . . . . . . . 8  |-  ( E. z  z  e.  U_ x  e.  A  B  <->  E. z  z  e.  {
z  |  E. x  e.  A  z  e.  B } )
1812, 14, 173bitr3i 266 . . . . . . 7  |-  ( -. 
{ y  |  E. x  e.  A  y  e.  B }  =  (/)  <->  E. z  z  e.  { z  |  E. x  e.  A  z  e.  B } )
1911, 18xchbinx 301 . . . . . 6  |-  ( { y  |  E. x  e.  A  y  e.  B }  =  (/)  <->  -.  E. z 
z  e.  { z  |  E. x  e.  A  z  e.  B } )
20 alnex 1533 . . . . . 6  |-  ( A. z  -.  z  e.  {
z  |  E. x  e.  A  z  e.  B }  <->  -.  E. z 
z  e.  { z  |  E. x  e.  A  z  e.  B } )
21 abid 2284 . . . . . . . 8  |-  ( z  e.  { z  |  E. x  e.  A  z  e.  B }  <->  E. x  e.  A  z  e.  B )
2221notbii 287 . . . . . . 7  |-  ( -.  z  e.  { z  |  E. x  e.  A  z  e.  B } 
<->  -.  E. x  e.  A  z  e.  B
)
2322albii 1556 . . . . . 6  |-  ( A. z  -.  z  e.  {
z  |  E. x  e.  A  z  e.  B }  <->  A. z  -.  E. x  e.  A  z  e.  B )
2419, 20, 233bitr2i 264 . . . . 5  |-  ( { y  |  E. x  e.  A  y  e.  B }  =  (/)  <->  A. z  -.  E. x  e.  A  z  e.  B )
2510, 24sylibr 203 . . . 4  |-  ( A  =  (/)  ->  { y  |  E. x  e.  A  y  e.  B }  =  (/) )
261, 25syl5eq 2340 . . 3  |-  ( A  =  (/)  ->  U_ x  e.  A  B  =  (/) )
27 0ss 3496 . . . 4  |-  (/)  C_  B
28 sseq1 3212 . . . 4  |-  ( U_ x  e.  A  B  =  (/)  ->  ( U_ x  e.  A  B  C_  B  <->  (/)  C_  B )
)
2927, 28mpbiri 224 . . 3  |-  ( U_ x  e.  A  B  =  (/)  ->  U_ x  e.  A  B  C_  B
)
3026, 29syl 15 . 2  |-  ( A  =  (/)  ->  U_ x  e.  A  B  C_  B
)
31 iunconst 3929 . . 3  |-  ( A  =/=  (/)  ->  U_ x  e.  A  B  =  B )
32 eqimss 3243 . . 3  |-  ( U_ x  e.  A  B  =  B  ->  U_ x  e.  A  B  C_  B
)
3331, 32syl 15 . 2  |-  ( A  =/=  (/)  ->  U_ x  e.  A  B  C_  B
)
3430, 33pm2.61ine 2535 1  |-  U_ x  e.  A  B  C_  B
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   E.wrex 2557    C_ wss 3165   (/)c0 3468   U_ciun 3921
This theorem is referenced by:  bnj1146  29139  bnj1145  29339  bnj1136  29343
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469  df-iun 3923
  Copyright terms: Public domain W3C validator