Theorem xp0 3457

Description: The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37.

Assertion

Ref	Expression
xp0	|- (A X. (/)) = (/)

Proof of Theorem xp0

Step	Hyp	Ref	Expression
1		xp0r 3234	. . 3 |- ((/) X. A) = (/)
2		cnveq 3287	. . 3 |- (((/) X. A) = (/) -> `'((/) X. A) = `'(/))
3	1, 2	ax-mp 7	. 2 |- `'((/) X. A) = `'(/)
4		cnvxp 3456	. 2 |- `'((/) X. A) = (A X. (/))
5		cnv0 3438	. 2 |- `'(/) = (/)
6	3, 4, 5	3eqtr3 1500	1 |- (A X. (/)) = (/)

Syntax hints:   = wceq 954  (/)c0 2276   X. cxp 3163  `'ccnv 3164

This theorem is referenced by:  xpnz 3458  xpdisj2 3461  dmxpss 3465  unixp 3509  fconst5 3839  aceq5lem3 4717  xpcdaen 4911  infxpidmlem4 7506  infxpdom 7522

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-xp 3179  df-rel 3180  df-cnv 3181

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