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Theorem bnj226 29038
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 2765 . 2  |-  A. x  e.  A  B  C_  C
3 iunss 4124 . 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 2697    C_ wss 3312   U_ciun 4085
This theorem is referenced by:  bnj229  29192  bnj1128  29296  bnj1145  29299
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-ral 2702  df-rex 2703  df-v 2950  df-in 3319  df-ss 3326  df-iun 4087
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