Theorem kmlem5 3584

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4.

Assertion

Ref	Expression
kmlem5	⊢ ((w ∈ x ∧ ¬ z = w) → ((z ∖ ∪(x ∖ {z})) ∩ (w ∖ ∪(x ∖ {w}))) = ∅)

Proof of Theorem kmlem5

Step	Hyp	Ref	Expression
1		difss 1596	. . . 4 ⊢ (w ∖ ∪(x ∖ {w})) ⊆ w
2		sslin 1662	. . . 4 ⊢ ((w ∖ ∪(x ∖ {w})) ⊆ w → ((z ∖ ∪(x ∖ {z})) ∩ (w ∖ ∪(x ∖ {w}))) ⊆ ((z ∖ ∪(x ∖ {z})) ∩ w))
3	1, 2	ax-mp 6	. . 3 ⊢ ((z ∖ ∪(x ∖ {z})) ∩ (w ∖ ∪(x ∖ {w}))) ⊆ ((z ∖ ∪(x ∖ {z})) ∩ w)
4		kmlem4 3583	. . . 4 ⊢ ((w ∈ x ∧ ¬ z = w) → ((z ∖ ∪(x ∖ {z})) ∩ w) = ∅)
5	4	sseq2d 1528	. . 3 ⊢ ((w ∈ x ∧ ¬ z = w) → (((z ∖ ∪(x ∖ {z})) ∩ (w ∖ ∪(x ∖ {w}))) ⊆ ((z ∖ ∪(x ∖ {z})) ∩ w) ↔ ((z ∖ ∪(x ∖ {z})) ∩ (w ∖ ∪(x ∖ {w}))) ⊆ ∅))
6	3, 5	mpbii 168	. 2 ⊢ ((w ∈ x ∧ ¬ z = w) → ((z ∖ ∪(x ∖ {z})) ∩ (w ∖ ∪(x ∖ {w}))) ⊆ ∅)
7		ss0b 1726	. 2 ⊢ (((z ∖ ∪(x ∖ {z})) ∩ (w ∖ ∪(x ∖ {w}))) ⊆ ∅ ↔ ((z ∖ ∪(x ∖ {z})) ∩ (w ∖ ∪(x ∖ {w}))) = ∅)
8	6, 7	sylib 173	1 ⊢ ((w ∈ x ∧ ¬ z = w) → ((z ∖ ∪(x ∖ {z})) ∩ (w ∖ ∪(x ∖ {w}))) = ∅)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = weq 797   ∈ wel 803   = wceq 1091   ∖ cdif 1484   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707  {csn 1808  ∪cuni 1919

This theorem is referenced by:  kmlem8 3587

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-uni 1920

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