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Theorem kmlem5 10196
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 4135 . . . 4 (𝑤 (𝑥 ∖ {𝑤})) ⊆ 𝑤
2 sslin 4242 . . . 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 10195 . . 3 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)
53, 4sseqtrid 4025 . 2 ((𝑤𝑥𝑧𝑤) → ((𝑧 (𝑥 ∖ {𝑧})) ∩ (𝑤 (𝑥 ∖ {𝑤}))) ⊆ ∅)
6 ss0b 4400 . 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 1539  wne 2939  cdif 3947  cin 3949  wss 3950  c0 4332  {csn 4625   cuni 4906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-in 3957  df-ss 3967  df-nul 4333  df-sn 4626  df-uni 4907
This theorem is referenced by:  kmlem9  10200
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