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Theorem un00 3586
 Description: Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
un00 ((A = B = ) ↔ (AB) = )

Proof of Theorem un00
StepHypRef Expression
1 uneq12 3413 . . 3 ((A = B = ) → (AB) = ())
2 un0 3575 . . 3 () =
31, 2syl6eq 2401 . 2 ((A = B = ) → (AB) = )
4 ssun1 3426 . . . . 5 A (AB)
5 sseq2 3293 . . . . 5 ((AB) = → (A (AB) ↔ A ))
64, 5mpbii 202 . . . 4 ((AB) = A )
7 ss0b 3580 . . . 4 (A A = )
86, 7sylib 188 . . 3 ((AB) = A = )
9 ssun2 3427 . . . . 5 B (AB)
10 sseq2 3293 . . . . 5 ((AB) = → (B (AB) ↔ B ))
119, 10mpbii 202 . . . 4 ((AB) = B )
12 ss0b 3580 . . . 4 (B B = )
1311, 12sylib 188 . . 3 ((AB) = B = )
148, 13jca 518 . 2 ((AB) = → (A = B = ))
153, 14impbii 180 1 ((A = B = ) ↔ (AB) = )
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∪ cun 3207   ⊆ wss 3257  ∅c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551 This theorem is referenced by:  undisj1  3602  undisj2  3603  0nelsuc  4400  addcass  4415
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