Theorem kmlem5 9580

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 4108	. . . 4 ⊢ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤})) ⊆ 𝑤
2		sslin 4211	. . . 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 9579	. . 3 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)
5	3, 4	sseqtrid 4019	. 2 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) ⊆ ∅)
6		ss0b 4351	. 2 ⊢ (((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) ⊆ ∅ ↔ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) = ∅)
7	5, 6	sylib 220	1 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) = ∅)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ≠ wne 3016   ∖ cdif 3933   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  {csn 4567  ∪ cuni 4838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3939  df-in 3943  df-ss 3952  df-nul 4292  df-sn 4568  df-uni 4839

This theorem is referenced by:  kmlem9  9584