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Theorem bnj1146 28502
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 1626 . . . . . 6  |-  F/ y ( x  e.  A  /\  w  e.  B
)
2 bnj1146.1 . . . . . . . 8  |-  ( y  e.  A  ->  A. x  y  e.  A )
32nfi 1557 . . . . . . 7  |-  F/ x  y  e.  A
4 nfv 1626 . . . . . . 7  |-  F/ x  w  e.  B
53, 4nfan 1836 . . . . . 6  |-  F/ x
( y  e.  A  /\  w  e.  B
)
6 eleq1 2449 . . . . . . 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 2025 . . . . 5  |-  ( E. x ( x  e.  A  /\  w  e.  B )  <->  E. y
( y  e.  A  /\  w  e.  B
) )
9 df-rex 2657 . . . . 5  |-  ( E. x  e.  A  w  e.  B  <->  E. x
( x  e.  A  /\  w  e.  B
) )
10 df-rex 2657 . . . . 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 2501 . . 3  |-  { w  |  E. x  e.  A  w  e.  B }  =  { w  |  E. y  e.  A  w  e.  B }
13 df-iun 4039 . . 3  |-  U_ x  e.  A  B  =  { w  |  E. x  e.  A  w  e.  B }
14 df-iun 4039 . . 3  |-  U_ y  e.  A  B  =  { w  |  E. y  e.  A  w  e.  B }
1512, 13, 143eqtr4i 2419 . 2  |-  U_ x  e.  A  B  =  U_ y  e.  A  B
16 bnj1143 28501 . 2  |-  U_ y  e.  A  B  C_  B
1715, 16eqsstri 3323 1  |-  U_ x  e.  A  B  C_  B
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546   E.wex 1547    e. wcel 1717   {cab 2375   E.wrex 2652    C_ wss 3265   U_ciun 4037
This theorem is referenced by:  bnj1145  28702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-v 2903  df-dif 3268  df-in 3272  df-ss 3279  df-nul 3574  df-iun 4039
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