Theorem kmlem5 10157

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	⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) = ∅)

Distinct variable group:   𝑥,𝑤,𝑧

Proof of Theorem kmlem5

Step	Hyp	Ref	Expression
1		difss 4093	. . . 4 ⊢ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤})) ⊆ 𝑤
2		sslin 4198	. . . 4 ⊢ ((𝑤 ∖ ∪ (𝑥 ∖ {𝑤})) ⊆ 𝑤 → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) ⊆ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤))
3	1, 2	ax-mp 5	. . 3 ⊢ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) ⊆ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤)
4		kmlem4 10156	. . 3 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)
5	3, 4	sseqtrid 3982	. 2 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) ⊆ ∅)
6		ss0b 4361	. 2 ⊢ (((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) ⊆ ∅ ↔ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) = ∅)
7	5, 6	sylib 221	1 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) = ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ≠ wne 2961   ∖ cdif 3905   ∩ cin 3907   ⊆ wss 3908  ∅c0 4289  {csn 4594  ∪ cuni 4877

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rab 3420  df-v 3460  df-dif 3911  df-in 3915  df-ss 3925  df-nul 4290  df-sn 4595  df-uni 4878

This theorem is used by:  kmlem9  10161