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Theorem bnj1143 28822
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 3907 . . . 4  |-  U_ x  e.  A  B  =  { y  |  E. x  e.  A  y  e.  B }
2 notnot 282 . . . . . . . 8  |-  ( A  =  (/)  <->  -.  -.  A  =  (/) )
3 neq0 3465 . . . . . . . 8  |-  ( -.  A  =  (/)  <->  E. x  x  e.  A )
42, 3xchbinx 301 . . . . . . 7  |-  ( A  =  (/)  <->  -.  E. x  x  e.  A )
5 df-rex 2549 . . . . . . . . 9  |-  ( E. x  e.  A  z  e.  B  <->  E. x
( x  e.  A  /\  z  e.  B
) )
6 exsimpl 1579 . . . . . . . . 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 1617 . . . . 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 3465 . . . . . . . 8  |-  ( -. 
U_ x  e.  A  B  =  (/)  <->  E. z 
z  e.  U_ x  e.  A  B )
131eqeq1i 2290 . . . . . . . . 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 3907 . . . . . . . . . 10  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
1615eleq2i 2347 . . . . . . . . 9  |-  ( z  e.  U_ x  e.  A  B  <->  z  e.  { z  |  E. x  e.  A  z  e.  B } )
1716exbii 1569 . . . . . . . 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 1530 . . . . . 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 2271 . . . . . . . 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 1553 . . . . . 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 2327 . . 3  |-  ( A  =  (/)  ->  U_ x  e.  A  B  =  (/) )
27 0ss 3483 . . . 4  |-  (/)  C_  B
28 sseq1 3199 . . . 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 3913 . . 3  |-  ( A  =/=  (/)  ->  U_ x  e.  A  B  =  B )
32 eqimss 3230 . . 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 2522 1  |-  U_ x  e.  A  B  C_  B
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   E.wrex 2544    C_ wss 3152   (/)c0 3455   U_ciun 3905
This theorem is referenced by:  bnj1146  28823  bnj1145  29023  bnj1136  29027
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-iun 3907
  Copyright terms: Public domain W3C validator