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Theorem un00 4271
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 4019 . . 3 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝐵) = (∅ ∪ ∅))
2 un0 4225 . . 3 (∅ ∪ ∅) = ∅
31, 2syl6eq 2824 . 2 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝐵) = ∅)
4 ssun1 4033 . . . . 5 𝐴 ⊆ (𝐴𝐵)
5 sseq2 3879 . . . . 5 ((𝐴𝐵) = ∅ → (𝐴 ⊆ (𝐴𝐵) ↔ 𝐴 ⊆ ∅))
64, 5mpbii 225 . . . 4 ((𝐴𝐵) = ∅ → 𝐴 ⊆ ∅)
7 ss0b 4231 . . . 4 (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
86, 7sylib 210 . . 3 ((𝐴𝐵) = ∅ → 𝐴 = ∅)
9 ssun2 4034 . . . . 5 𝐵 ⊆ (𝐴𝐵)
10 sseq2 3879 . . . . 5 ((𝐴𝐵) = ∅ → (𝐵 ⊆ (𝐴𝐵) ↔ 𝐵 ⊆ ∅))
119, 10mpbii 225 . . . 4 ((𝐴𝐵) = ∅ → 𝐵 ⊆ ∅)
12 ss0b 4231 . . . 4 (𝐵 ⊆ ∅ ↔ 𝐵 = ∅)
1311, 12sylib 210 . . 3 ((𝐴𝐵) = ∅ → 𝐵 = ∅)
148, 13jca 504 . 2 ((𝐴𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅))
153, 14impbii 201 1 ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387   = wceq 1507  cun 3823  wss 3825  c0 4173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174
This theorem is referenced by:  undisj1  4288  undisj2  4289  disjpr2  4517  rankxplim3  9096  ssxr  10502  rpnnen2lem12  15428  wwlksnext  27371  asindmre  34366  iunrelexp0  39355  uneqsn  39681
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