Theorem unixpss 3258

Description: The double class union of a cross product is included in the union of its arguments.

Assertion

Ref	Expression
unixpss	|- U.U.(A X. B) (_ (A u. B)

Proof of Theorem unixpss

Step	Hyp	Ref	Expression
1		xpsspw 3257	. . . . 5 |- (A X. B) (_ P~P~(A u. B)
2		uniss 2521	. . . . 5 |- ((A X. B) (_ P~P~(A u. B) -> U.(A X. B) (_ U.P~P~(A u. B))
3	1, 2	ax-mp 7	. . . 4 |- U.(A X. B) (_ U.P~P~(A u. B)
4		unipw 2756	. . . 4 |- U.P~P~(A u. B) = P~(A u. B)
5	3, 4	sseqtr 2093	. . 3 |- U.(A X. B) (_ P~(A u. B)
6		uniss 2521	. . 3 |- (U.(A X. B) (_ P~(A u. B) -> U.U.(A X. B) (_ U.P~(A u. B))
7	5, 6	ax-mp 7	. 2 |- U.U.(A X. B) (_ U.P~(A u. B)
8		unipw 2756	. 2 |- U.P~(A u. B) = (A u. B)
9	7, 8	sseqtr 2093	1 |- U.U.(A X. B) (_ (A u. B)

Syntax hints:   u. cun 2045   (_ wss 2047  P~cpw 2401  U.cuni 2503   X. cxp 3168

This theorem is referenced by:  relfld 3515

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-opab 2667  df-xp 3184  df-rel 3185

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