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Theorem bnj226 29544
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 2787 . 2  |-  A. x  e.  A  B  C_  C
3 iunss 4338 . 2  |-  ( U_ x  e.  A  B  C_  C  <->  A. x  e.  A  B  C_  C )
42, 3mpbir 213 1  |-  U_ x  e.  A  B  C_  C
Colors of variables: wff setvar class
Syntax hints:   A.wral 2776    C_ wss 3437   U_ciun 4297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-rex 2782  df-v 3084  df-in 3444  df-ss 3451  df-iun 4299
This theorem is referenced by:  bnj229  29697  bnj1128  29801  bnj1145  29804
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