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Theorem bnj226 29078
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 2623 . 2  |-  A. x  e.  A  B  C_  C
3 iunss 3959 . 2  |-  ( U_ x  e.  A  B  C_  C  <->  A. x  e.  A  B  C_  C )
42, 3mpbir 200 1  |-  U_ x  e.  A  B  C_  C
Colors of variables: wff set class
Syntax hints:   A.wral 2556    C_ wss 3165   U_ciun 3921
This theorem is referenced by:  bnj229  29232  bnj1128  29336  bnj1145  29339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-iun 3923
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