Theorem bnj1146 29675

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

Hypothesis

Ref	Expression
bnj1146.1	|- ( y e. A -> A. x y e. A )

Assertion

Ref	Expression
bnj1146	|- U_ x e. A B C_ B

Distinct variable groups:   y, A   x, B, y

Allowed substitution hint:   A( x)

Proof of Theorem bnj1146

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1769	. . . . . 6 |- F/ y ( x e. A /\ w e. B )
2		bnj1146.1	. . . . . . . 8 |- ( y e. A -> A. x y e. A )
3	2	nfi 1682	. . . . . . 7 |- F/ x y e. A
4		nfv 1769	. . . . . . 7 |- F/ x w e. B
5	3, 4	nfan 2031	. . . . . 6 |- F/ x ( y e. A /\ w e. B )
6		eleq1 2537	. . . . . . 7 |- ( x = y -> ( x e. A <-> y e. A ) )
7	6	anbi1d 719	. . . . . 6 |- ( x = y -> ( ( x e. A /\ w e. B ) <-> ( y e. A /\ w e. B ) ) )
8	1, 5, 7	cbvex 2128	. . . . 5 |- ( E. x ( x e. A /\ w e. B ) <-> E. y ( y e. A /\ w e. B ) )
9		df-rex 2762	. . . . 5 |- ( E. x e. A w e. B <-> E. x ( x e. A /\ w e. B ) )
10		df-rex 2762	. . . . 5 |- ( E. y e. A w e. B <-> E. y ( y e. A /\ w e. B ) )
11	8, 9, 10	3bitr4i 285	. . . 4 |- ( E. x e. A w e. B <-> E. y e. A w e. B )
12	11	abbii 2587	. . 3 |- { w | E. x e. A w e. B } = { w | E. y e. A w e. B }
13		df-iun 4271	. . 3 |- U_ x e. A B = { w | E. x e. A w e. B }
14		df-iun 4271	. . 3 |- U_ y e. A B = { w | E. y e. A w e. B }
15	12, 13, 14	3eqtr4i 2503	. 2 |- U_ x e. A B = U_ y e. A B
16		bnj1143 29674	. 2 |- U_ y e. A B C_ B
17	15, 16	eqsstri 3448	1 |- U_ x e. A B C_ B

Syntax hints:   -> wi 4   /\ wa 376   A.wal 1450   E.wex 1671   e. wcel 1904   {cab 2457   E.wrex 2757   C_ wss 3390   U_ciun 4269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-iun 4271

This theorem is referenced by:  bnj1145  29874