Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  un00 GIF version

Theorem un00 3263
 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 3092 . . 3 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝐵) = (∅ ∪ ∅))
2 un0 3251 . . 3 (∅ ∪ ∅) = ∅
31, 2syl6eq 2088 . 2 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝐵) = ∅)
4 ssun1 3106 . . . . 5 𝐴 ⊆ (𝐴𝐵)
5 sseq2 2967 . . . . 5 ((𝐴𝐵) = ∅ → (𝐴 ⊆ (𝐴𝐵) ↔ 𝐴 ⊆ ∅))
64, 5mpbii 136 . . . 4 ((𝐴𝐵) = ∅ → 𝐴 ⊆ ∅)
7 ss0b 3256 . . . 4 (𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
86, 7sylib 127 . . 3 ((𝐴𝐵) = ∅ → 𝐴 = ∅)
9 ssun2 3107 . . . . 5 𝐵 ⊆ (𝐴𝐵)
10 sseq2 2967 . . . . 5 ((𝐴𝐵) = ∅ → (𝐵 ⊆ (𝐴𝐵) ↔ 𝐵 ⊆ ∅))
119, 10mpbii 136 . . . 4 ((𝐴𝐵) = ∅ → 𝐵 ⊆ ∅)
12 ss0b 3256 . . . 4 (𝐵 ⊆ ∅ ↔ 𝐵 = ∅)
1311, 12sylib 127 . . 3 ((𝐴𝐵) = ∅ → 𝐵 = ∅)
148, 13jca 290 . 2 ((𝐴𝐵) = ∅ → (𝐴 = ∅ ∧ 𝐵 = ∅))
153, 14impbii 117 1 ((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴𝐵) = ∅)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1243   ∪ cun 2915   ⊆ wss 2917  ∅c0 3224 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225 This theorem is referenced by:  undisj1  3279  undisj2  3280  disjpr2  3434
 Copyright terms: Public domain W3C validator