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Theorem bnj226 28441
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj226.1  |-  B  C_  C
Assertion
Ref Expression
bnj226  |-  U_ x  e.  A  B  C_  C
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem bnj226
StepHypRef Expression
1 bnj226.1 . . 3  |-  B  C_  C
21rgenw 2718 . 2  |-  A. x  e.  A  B  C_  C
3 iunss 4075 . 2  |-  ( U_ x  e.  A  B  C_  C  <->  A. x  e.  A  B  C_  C )
42, 3mpbir 201 1  |-  U_ x  e.  A  B  C_  C
Colors of variables: wff set class
Syntax hints:   A.wral 2651    C_ wss 3265   U_ciun 4037
This theorem is referenced by:  bnj229  28595  bnj1128  28699  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-ral 2656  df-rex 2657  df-v 2903  df-in 3272  df-ss 3279  df-iun 4039
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