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Theorem un00 3329
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 3149 . . 3 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝐵) = (∅ ∪ ∅))
2 un0 3316 . . 3 (∅ ∪ ∅) = ∅
31, 2syl6eq 2136 . 2 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝐵) = ∅)
4 ssun1 3163 . . . . 5 𝐴 ⊆ (𝐴𝐵)
5 sseq2 3048 . . . . 5 ((𝐴𝐵) = ∅ → (𝐴 ⊆ (𝐴𝐵) ↔ 𝐴 ⊆ ∅))
64, 5mpbii 146 . . . 4 ((𝐴𝐵) = ∅ → 𝐴 ⊆ ∅)
7 ss0b 3322 . . . 4 (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
86, 7sylib 120 . . 3 ((𝐴𝐵) = ∅ → 𝐴 = ∅)
9 ssun2 3164 . . . . 5 𝐵 ⊆ (𝐴𝐵)
10 sseq2 3048 . . . . 5 ((𝐴𝐵) = ∅ → (𝐵 ⊆ (𝐴𝐵) ↔ 𝐵 ⊆ ∅))
119, 10mpbii 146 . . . 4 ((𝐴𝐵) = ∅ → 𝐵 ⊆ ∅)
12 ss0b 3322 . . . 4 (𝐵 ⊆ ∅ ↔ 𝐵 = ∅)
1311, 12sylib 120 . . 3 ((𝐴𝐵) = ∅ → 𝐵 = ∅)
148, 13jca 300 . 2 ((𝐴𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅))
153, 14impbii 124 1 ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴𝐵) = ∅)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1289  cun 2997  wss 2999  c0 3286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287
This theorem is referenced by:  undisj1  3340  undisj2  3341  disjpr2  3506
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