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Theorem kmlem5 10193
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 4146 . . . 4 (𝑤 (𝑥 ∖ {𝑤})) ⊆ 𝑤
2 sslin 4251 . . . 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 10192 . . 3 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)
53, 4sseqtrid 4048 . 2 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) ⊆ ∅)
6 ss0b 4407 . 2 (((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) ⊆ ∅ ↔ ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) = ∅)
75, 6sylib 218 1 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wne 2938  cdif 3960  cin 3962  wss 3963  c0 4339  {csn 4631   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-uni 4913
This theorem is referenced by:  kmlem9  10197
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