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Theorem kmlem5 10171
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 4127 . . . 4 (𝑤 (𝑥 ∖ {𝑤})) ⊆ 𝑤
2 sslin 4230 . . . 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 10170 . . 3 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)
53, 4sseqtrid 4030 . 2 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) ⊆ ∅)
6 ss0b 4393 . 2 (((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) ⊆ ∅ ↔ ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) = ∅)
75, 6sylib 217 1 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wne 2935  cdif 3941  cin 3943  wss 3944  c0 4318  {csn 4624   cuni 4903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rab 3428  df-v 3471  df-dif 3947  df-in 3951  df-ss 3961  df-nul 4319  df-sn 4625  df-uni 4904
This theorem is referenced by:  kmlem9  10175
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