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Theorem kmlem5 9014
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 3770 . . . 4 (𝑤 (𝑥 ∖ {𝑤})) ⊆ 𝑤
2 sslin 3872 . . . 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 9013 . . 3 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)
53, 4syl5sseq 3686 . 2 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) ⊆ ∅)
6 ss0b 4006 . 2 (((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) ⊆ ∅ ↔ ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) = ∅)
75, 6sylib 208 1 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wne 2823  cdif 3604  cin 3606  wss 3607  c0 3948  {csn 4210   cuni 4468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-uni 4469
This theorem is referenced by:  kmlem9  9018
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