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Theorem xpexr 3471
Description: If a cross product is a set, one of its components must be a set.
Assertion
Ref Expression
xpexr |- ((A X. B) e. C -> (A e. V \/ B e. V))
Proof of Theorem xpexr
StepHypRef Expression
1 0ex 2706 . . . . . 6 |- (/) e. V
2 eleq1 1531 . . . . . 6 |- (A = (/) -> (A e. V <-> (/) e. V))
31, 2mpbiri 194 . . . . 5 |- (A = (/) -> A e. V)
43pm2.24d 105 . . . 4 |- (A = (/) -> (-. A e. V -> B e. V))
54a1d 12 . . 3 |- (A = (/) -> ((A X. B) e. C -> (-. A e. V -> B e. V)))
6 rnxp 3464 . . . . . 6 |- (A =/= (/) -> ran ( A X. B) = B)
76eleq1d 1537 . . . . 5 |- (A =/= (/) -> (ran ( A X. B) e. V <-> B e. V))
8 rnexg 3353 . . . . 5 |- ((A X. B) e. C -> ran ( A X. B) e. V)
97, 8syl5bi 208 . . . 4 |- (A =/= (/) -> ((A X. B) e. C -> B e. V))
109a1dd 42 . . 3 |- (A =/= (/) -> ((A X. B) e. C -> (-. A e. V -> B e. V)))
115, 10pm2.61ine 1631 . 2 |- ((A X. B) e. C -> (-. A e. V -> B e. V))
1211orrd 233 1 |- ((A X. B) e. C -> (A e. V \/ B e. V))
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   = wceq 954   e. wcel 956   =/= wne 1582  Vcvv 1807  (/)c0 2276   X. cxp 3163  ran crn 3166
This theorem is referenced by:  ismsg 7750
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-9 963  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-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861
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-ral 1646  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-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-rel 3180  df-cnv 3181  df-dm 3183  df-rn 3184
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