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Theorem un00 3291
 Description: Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
un00

Proof of Theorem un00
StepHypRef Expression
1 uneq12 3120 . . 3
2 un0 3279 . . 3
31, 2syl6eq 2104 . 2
4 ssun1 3134 . . . . 5
5 sseq2 2995 . . . . 5
64, 5mpbii 140 . . . 4
7 ss0b 3284 . . . 4
86, 7sylib 131 . . 3
9 ssun2 3135 . . . . 5
10 sseq2 2995 . . . . 5
119, 10mpbii 140 . . . 4
12 ss0b 3284 . . . 4
1311, 12sylib 131 . . 3
148, 13jca 294 . 2
153, 14impbii 121 1
 Colors of variables: wff set class Syntax hints:   wa 101   wb 102   wceq 1259   cun 2943   wss 2945  c0 3252 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253 This theorem is referenced by:  undisj1  3307  undisj2  3308  disjpr2  3462
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