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Theorem bnj1146 29675
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1146.1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Assertion
Ref Expression
bnj1146  |-  U_ x  e.  A  B  C_  B
Distinct variable groups:    y, A    x, B, y
Allowed substitution hint:    A( x)

Proof of Theorem bnj1146
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfv 1769 . . . . . 6  |-  F/ y ( x  e.  A  /\  w  e.  B
)
2 bnj1146.1 . . . . . . . 8  |-  ( y  e.  A  ->  A. x  y  e.  A )
32nfi 1682 . . . . . . 7  |-  F/ x  y  e.  A
4 nfv 1769 . . . . . . 7  |-  F/ x  w  e.  B
53, 4nfan 2031 . . . . . 6  |-  F/ x
( y  e.  A  /\  w  e.  B
)
6 eleq1 2537 . . . . . . 7  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
76anbi1d 719 . . . . . 6  |-  ( x  =  y  ->  (
( x  e.  A  /\  w  e.  B
)  <->  ( y  e.  A  /\  w  e.  B ) ) )
81, 5, 7cbvex 2128 . . . . 5  |-  ( E. x ( x  e.  A  /\  w  e.  B )  <->  E. y
( y  e.  A  /\  w  e.  B
) )
9 df-rex 2762 . . . . 5  |-  ( E. x  e.  A  w  e.  B  <->  E. x
( x  e.  A  /\  w  e.  B
) )
10 df-rex 2762 . . . . 5  |-  ( E. y  e.  A  w  e.  B  <->  E. y
( y  e.  A  /\  w  e.  B
) )
118, 9, 103bitr4i 285 . . . 4  |-  ( E. x  e.  A  w  e.  B  <->  E. y  e.  A  w  e.  B )
1211abbii 2587 . . 3  |-  { w  |  E. x  e.  A  w  e.  B }  =  { w  |  E. y  e.  A  w  e.  B }
13 df-iun 4271 . . 3  |-  U_ x  e.  A  B  =  { w  |  E. x  e.  A  w  e.  B }
14 df-iun 4271 . . 3  |-  U_ y  e.  A  B  =  { w  |  E. y  e.  A  w  e.  B }
1512, 13, 143eqtr4i 2503 . 2  |-  U_ x  e.  A  B  =  U_ y  e.  A  B
16 bnj1143 29674 . 2  |-  U_ y  e.  A  B  C_  B
1715, 16eqsstri 3448 1  |-  U_ x  e.  A  B  C_  B
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376   A.wal 1450   E.wex 1671    e. wcel 1904   {cab 2457   E.wrex 2757    C_ wss 3390   U_ciun 4269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-iun 4271
This theorem is referenced by:  bnj1145  29874
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