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Theorem bnj1146 29089
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 1629 . . . . . 6  |-  F/ y ( x  e.  A  /\  w  e.  B
)
2 bnj1146.1 . . . . . . . 8  |-  ( y  e.  A  ->  A. x  y  e.  A )
32nfi 1560 . . . . . . 7  |-  F/ x  y  e.  A
4 nfv 1629 . . . . . . 7  |-  F/ x  w  e.  B
53, 4nfan 1846 . . . . . 6  |-  F/ x
( y  e.  A  /\  w  e.  B
)
6 eleq1 2495 . . . . . . 7  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
76anbi1d 686 . . . . . 6  |-  ( x  =  y  ->  (
( x  e.  A  /\  w  e.  B
)  <->  ( y  e.  A  /\  w  e.  B ) ) )
81, 5, 7cbvex 1983 . . . . 5  |-  ( E. x ( x  e.  A  /\  w  e.  B )  <->  E. y
( y  e.  A  /\  w  e.  B
) )
9 df-rex 2703 . . . . 5  |-  ( E. x  e.  A  w  e.  B  <->  E. x
( x  e.  A  /\  w  e.  B
) )
10 df-rex 2703 . . . . 5  |-  ( E. y  e.  A  w  e.  B  <->  E. y
( y  e.  A  /\  w  e.  B
) )
118, 9, 103bitr4i 269 . . . 4  |-  ( E. x  e.  A  w  e.  B  <->  E. y  e.  A  w  e.  B )
1211abbii 2547 . . 3  |-  { w  |  E. x  e.  A  w  e.  B }  =  { w  |  E. y  e.  A  w  e.  B }
13 df-iun 4087 . . 3  |-  U_ x  e.  A  B  =  { w  |  E. x  e.  A  w  e.  B }
14 df-iun 4087 . . 3  |-  U_ y  e.  A  B  =  { w  |  E. y  e.  A  w  e.  B }
1512, 13, 143eqtr4i 2465 . 2  |-  U_ x  e.  A  B  =  U_ y  e.  A  B
16 bnj1143 29088 . 2  |-  U_ y  e.  A  B  C_  B
1715, 16eqsstri 3370 1  |-  U_ x  e.  A  B  C_  B
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550    e. wcel 1725   {cab 2421   E.wrex 2698    C_ wss 3312   U_ciun 4085
This theorem is referenced by:  bnj1145  29289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-iun 4087
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