Theorem bnj226 30802

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

Step	Hyp	Ref	Expression
1		bnj226.1	. . 3 |- B C_ C
2	1	rgenw 2924	. 2 |- A. x e. A B C_ C
3		iunss 4561	. 2 |- ( U_ x e. A B C_ C <-> A. x e. A B C_ C )
4	2, 3	mpbir 221	1 |- U_ x e. A B C_ C

Syntax hints:   A.wral 2912   C_ wss 3574   U_ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-iun 4522

This theorem is referenced by:  bnj229  30954  bnj1128  31058  bnj1145  31061