Theorem kmlem5 10075

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 4073	. . . 4 ⊢ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤})) ⊆ 𝑤
2		sslin 4178	. . . 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 10074	. . 3 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ 𝑤) = ∅)
5	3, 4	sseqtrid 3964	. 2 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) ⊆ ∅)
6		ss0b 4336	. 2 ⊢ (((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) ⊆ ∅ ↔ ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) = ∅)
7	5, 6	sylib 219	1 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑧 ≠ 𝑤) → ((𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ∩ (𝑤 ∖ ∪ (𝑥 ∖ {𝑤}))) = ∅)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ≠ wne 2935   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268  {csn 4562  ∪ cuni 4845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rab 3393  df-v 3434  df-dif 3893  df-in 3897  df-ss 3907  df-nul 4269  df-sn 4563  df-uni 4846

This theorem is referenced by:  kmlem9  10079