Theorem bnj1146 30862

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 1843	. . . . . 6 |- F/ y ( x e. A /\ w e. B )
2		bnj1146.1	. . . . . . . 8 |- ( y e. A -> A. x y e. A )
3	2	nf5i 2024	. . . . . . 7 |- F/ x y e. A
4		nfv 1843	. . . . . . 7 |- F/ x w e. B
5	3, 4	nfan 1828	. . . . . 6 |- F/ x ( y e. A /\ w e. B )
6		eleq1 2689	. . . . . . 7 |- ( x = y -> ( x e. A <-> y e. A ) )
7	6	anbi1d 741	. . . . . 6 |- ( x = y -> ( ( x e. A /\ w e. B ) <-> ( y e. A /\ w e. B ) ) )
8	1, 5, 7	cbvex 2272	. . . . 5 |- ( E. x ( x e. A /\ w e. B ) <-> E. y ( y e. A /\ w e. B ) )
9		df-rex 2918	. . . . 5 |- ( E. x e. A w e. B <-> E. x ( x e. A /\ w e. B ) )
10		df-rex 2918	. . . . 5 |- ( E. y e. A w e. B <-> E. y ( y e. A /\ w e. B ) )
11	8, 9, 10	3bitr4i 292	. . . 4 |- ( E. x e. A w e. B <-> E. y e. A w e. B )
12	11	abbii 2739	. . 3 |- { w | E. x e. A w e. B } = { w | E. y e. A w e. B }
13		df-iun 4522	. . 3 |- U_ x e. A B = { w | E. x e. A w e. B }
14		df-iun 4522	. . 3 |- U_ y e. A B = { w | E. y e. A w e. B }
15	12, 13, 14	3eqtr4i 2654	. 2 |- U_ x e. A B = U_ y e. A B
16		bnj1143 30861	. 2 |- U_ y e. A B C_ B
17	15, 16	eqsstri 3635	1 |- U_ x e. A B C_ B

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   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-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-iun 4522

This theorem is referenced by:  bnj1145  31061