Theorem bnj1149 30863

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1149.1	|- ( ph -> A e. _V )
bnj1149.2	|- ( ph -> B e. _V )

Assertion

Ref	Expression
bnj1149	|- ( ph -> ( A u. B ) e. _V )

Proof of Theorem bnj1149

Step	Hyp	Ref	Expression
1		bnj1149.1	. 2 |- ( ph -> A e. _V )
2		bnj1149.2	. 2 |- ( ph -> B e. _V )
3		unexg 6959	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A u. B ) e. _V )
4	1, 2, 3	syl2anc 693	1 |- ( ph -> ( A u. B ) e. _V )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   u. cun 3572

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

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-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by:  bnj1136  31065  bnj1413  31103  bnj1452  31120  bnj1489  31124