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Theorem un00 4153
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 3911 . . 3 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝐵) = (∅ ∪ ∅))
2 un0 4109 . . 3 (∅ ∪ ∅) = ∅
31, 2syl6eq 2820 . 2 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝐵) = ∅)
4 ssun1 3925 . . . . 5 𝐴 ⊆ (𝐴𝐵)
5 sseq2 3774 . . . . 5 ((𝐴𝐵) = ∅ → (𝐴 ⊆ (𝐴𝐵) ↔ 𝐴 ⊆ ∅))
64, 5mpbii 223 . . . 4 ((𝐴𝐵) = ∅ → 𝐴 ⊆ ∅)
7 ss0b 4115 . . . 4 (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
86, 7sylib 208 . . 3 ((𝐴𝐵) = ∅ → 𝐴 = ∅)
9 ssun2 3926 . . . . 5 𝐵 ⊆ (𝐴𝐵)
10 sseq2 3774 . . . . 5 ((𝐴𝐵) = ∅ → (𝐵 ⊆ (𝐴𝐵) ↔ 𝐵 ⊆ ∅))
119, 10mpbii 223 . . . 4 ((𝐴𝐵) = ∅ → 𝐵 ⊆ ∅)
12 ss0b 4115 . . . 4 (𝐵 ⊆ ∅ ↔ 𝐵 = ∅)
1311, 12sylib 208 . . 3 ((𝐴𝐵) = ∅ → 𝐵 = ∅)
148, 13jca 495 . 2 ((𝐴𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅))
153, 14impbii 199 1 ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1630  cun 3719  wss 3721  c0 4061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062
This theorem is referenced by:  undisj1  4170  undisj2  4171  disjpr2  4383  rankxplim3  8907  ssxr  10308  rpnnen2lem12  15159  wwlksnext  27035  asindmre  33820  iunrelexp0  38513  uneqsn  38840
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