Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  kmlem5 Structured version   Visualization version   GIF version

Theorem kmlem5 9372
 Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)
Assertion
Ref Expression
kmlem5 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) = ∅)
Distinct variable group:   𝑥,𝑤,𝑧

Proof of Theorem kmlem5
StepHypRef Expression
1 difss 3992 . . . 4 (𝑤 (𝑥 ∖ {𝑤})) ⊆ 𝑤
2 sslin 4092 . . . 4 ((𝑤 (𝑥 ∖ {𝑤})) ⊆ 𝑤 → ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) ⊆ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤))
31, 2ax-mp 5 . . 3 ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) ⊆ ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤)
4 kmlem4 9371 . . 3 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)
53, 4syl5sseq 3903 . 2 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) ⊆ ∅)
6 ss0b 4231 . 2 (((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) ⊆ ∅ ↔ ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) = ∅)
75, 6sylib 210 1 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ≠ wne 2961   ∖ cdif 3820   ∩ cin 3822   ⊆ wss 3823  ∅c0 4172  {csn 4435  ∪ cuni 4708 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 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744 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 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-v 3411  df-dif 3826  df-in 3830  df-ss 3837  df-nul 4173  df-sn 4436  df-uni 4709 This theorem is referenced by:  kmlem9  9376
 Copyright terms: Public domain W3C validator