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Theorem bnj226 32080
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 2901 . 2  |-  A. x  e.  A  B  C_  C
3 iunss 4322 . 2  |-  ( U_ x  e.  A  B  C_  C  <->  A. x  e.  A  B  C_  C )
42, 3mpbir 209 1  |-  U_ x  e.  A  B  C_  C
Colors of variables: wff setvar class
Syntax hints:   A.wral 2799    C_ wss 3439   U_ciun 4282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-v 3080  df-in 3446  df-ss 3453  df-iun 4284
This theorem is referenced by:  bnj229  32232  bnj1128  32336  bnj1145  32339
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