Theorem djussxp 4481

Description: Disjoint union is a subset of a cross product. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Assertion

Ref	Expression
djussxp	|- U_ x e. A ( { x } X. B ) C_ ( A X. _V )

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem djussxp

Step	Hyp	Ref	Expression
1		iunss 3698	. 2 |- ( U_ x e. A ( { x } X. B ) C_ ( A X. _V ) <-> A. x e. A ( { x } X. B ) C_ ( A X. _V ) )
2		snssi 3508	. . 3 |- ( x e. A -> { x } C_ A )
3		ssv 2965	. . 3 |- B C_ _V
4		xpss12 4445	. . 3 |- ( ( { x } C_ A /\ B C_ _V ) -> ( { x } X. B ) C_ ( A X. _V ) )
5	2, 3, 4	sylancl 392	. 2 |- ( x e. A -> ( { x } X. B ) C_ ( A X. _V ) )
6	1, 5	mprgbir 2379	1 |- U_ x e. A ( { x } X. B ) C_ ( A X. _V )

Syntax hints:   e. wcel 1393   _Vcvv 2557   C_ wss 2917   {csn 3375   U_ciun 3657   X. cxp 4343

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-sn 3381  df-iun 3659  df-opab 3819  df-xp 4351

This theorem is referenced by:  djudisj  4750