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Theorem un00 3257
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 3086 . . 3 ((A = ∅ B = ∅) → (AB) = (∅ ∪ ∅))
2 un0 3245 . . 3 (∅ ∪ ∅) = ∅
31, 2syl6eq 2085 . 2 ((A = ∅ B = ∅) → (AB) = ∅)
4 ssun1 3100 . . . . 5 A ⊆ (AB)
5 sseq2 2961 . . . . 5 ((AB) = ∅ → (A ⊆ (AB) ↔ A ⊆ ∅))
64, 5mpbii 136 . . . 4 ((AB) = ∅ → A ⊆ ∅)
7 ss0b 3250 . . . 4 (A ⊆ ∅ ↔ A = ∅)
86, 7sylib 127 . . 3 ((AB) = ∅ → A = ∅)
9 ssun2 3101 . . . . 5 B ⊆ (AB)
10 sseq2 2961 . . . . 5 ((AB) = ∅ → (B ⊆ (AB) ↔ B ⊆ ∅))
119, 10mpbii 136 . . . 4 ((AB) = ∅ → B ⊆ ∅)
12 ss0b 3250 . . . 4 (B ⊆ ∅ ↔ B = ∅)
1311, 12sylib 127 . . 3 ((AB) = ∅ → B = ∅)
148, 13jca 290 . 2 ((AB) = ∅ → (A = ∅ B = ∅))
153, 14impbii 117 1 ((A = ∅ B = ∅) ↔ (AB) = ∅)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  cun 2909  wss 2911  c0 3218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219
This theorem is referenced by:  undisj1  3273  undisj2  3274  disjpr2  3425
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