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Theorem un00 4343
 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 4066 . . 3 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝐵) = (∅ ∪ ∅))
2 un0 4290 . . 3 (∅ ∪ ∅) = ∅
31, 2eqtrdi 2810 . 2 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝐵) = ∅)
4 ssun1 4080 . . . . 5 𝐴 ⊆ (𝐴𝐵)
5 sseq2 3921 . . . . 5 ((𝐴𝐵) = ∅ → (𝐴 ⊆ (𝐴𝐵) ↔ 𝐴 ⊆ ∅))
64, 5mpbii 236 . . . 4 ((𝐴𝐵) = ∅ → 𝐴 ⊆ ∅)
7 ss0b 4297 . . . 4 (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
86, 7sylib 221 . . 3 ((𝐴𝐵) = ∅ → 𝐴 = ∅)
9 ssun2 4081 . . . . 5 𝐵 ⊆ (𝐴𝐵)
10 sseq2 3921 . . . . 5 ((𝐴𝐵) = ∅ → (𝐵 ⊆ (𝐴𝐵) ↔ 𝐵 ⊆ ∅))
119, 10mpbii 236 . . . 4 ((𝐴𝐵) = ∅ → 𝐵 ⊆ ∅)
12 ss0b 4297 . . . 4 (𝐵 ⊆ ∅ ↔ 𝐵 = ∅)
1311, 12sylib 221 . . 3 ((𝐴𝐵) = ∅ → 𝐵 = ∅)
148, 13jca 515 . 2 ((𝐴𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅))
153, 14impbii 212 1 ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴𝐵) = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1539   ∪ cun 3859   ⊆ wss 3861  ∅c0 4228 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-v 3412  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229 This theorem is referenced by:  undisj1  4362  undisj2  4363  disjpr2  4610  rankxplim3  9357  ssxr  10762  rpnnen2lem12  15640  wwlksnext  27793  asindmre  35456  iunrelexp0  40822  uneqsn  41145
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