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Theorem un00 4236
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 3984 . . 3 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝐵) = (∅ ∪ ∅))
2 un0 4192 . . 3 (∅ ∪ ∅) = ∅
31, 2syl6eq 2829 . 2 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝐵) = ∅)
4 ssun1 3998 . . . . 5 𝐴 ⊆ (𝐴𝐵)
5 sseq2 3845 . . . . 5 ((𝐴𝐵) = ∅ → (𝐴 ⊆ (𝐴𝐵) ↔ 𝐴 ⊆ ∅))
64, 5mpbii 225 . . . 4 ((𝐴𝐵) = ∅ → 𝐴 ⊆ ∅)
7 ss0b 4198 . . . 4 (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
86, 7sylib 210 . . 3 ((𝐴𝐵) = ∅ → 𝐴 = ∅)
9 ssun2 3999 . . . . 5 𝐵 ⊆ (𝐴𝐵)
10 sseq2 3845 . . . . 5 ((𝐴𝐵) = ∅ → (𝐵 ⊆ (𝐴𝐵) ↔ 𝐵 ⊆ ∅))
119, 10mpbii 225 . . . 4 ((𝐴𝐵) = ∅ → 𝐵 ⊆ ∅)
12 ss0b 4198 . . . 4 (𝐵 ⊆ ∅ ↔ 𝐵 = ∅)
1311, 12sylib 210 . . 3 ((𝐴𝐵) = ∅ → 𝐵 = ∅)
148, 13jca 507 . 2 ((𝐴𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅))
153, 14impbii 201 1 ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1601  cun 3789  wss 3791  c0 4140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141
This theorem is referenced by:  undisj1  4253  undisj2  4254  disjpr2  4479  rankxplim3  9041  ssxr  10446  rpnnen2lem12  15358  wwlksnext  27254  asindmre  34115  iunrelexp0  38944  uneqsn  39270
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