Theorem enadjlem1 6060

Description: Lemma for enadj 6061. Calculate equality of differences. (Contributed by SF, 25-Feb-2015.)

Assertion

Ref	Expression
enadjlem1	⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → (A ∖ {Y}) = (B ∖ {X}))

Proof of Theorem enadjlem1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elsni 3758	. . . . . . . . 9 ⊢ (x ∈ {Y} → x = Y)
2	1	necon3ai 2557	. . . . . . . 8 ⊢ (x ≠ Y → ¬ x ∈ {Y})
3	2	ad2antll 709	. . . . . . 7 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (x ∈ A ∧ x ≠ Y)) → ¬ x ∈ {Y})
4		ssun1 3427	. . . . . . . . . . 11 ⊢ A ⊆ (A ∪ {X})
5	4	sseli 3270	. . . . . . . . . 10 ⊢ (x ∈ A → x ∈ (A ∪ {X}))
6	5	ad2antrl 708	. . . . . . . . 9 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (x ∈ A ∧ x ≠ Y)) → x ∈ (A ∪ {X}))
7		simpl1 958	. . . . . . . . 9 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (x ∈ A ∧ x ≠ Y)) → (A ∪ {X}) = (B ∪ {Y}))
8	6, 7	eleqtrd 2429	. . . . . . . 8 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (x ∈ A ∧ x ≠ Y)) → x ∈ (B ∪ {Y}))
9		elun 3221	. . . . . . . 8 ⊢ (x ∈ (B ∪ {Y}) ↔ (x ∈ B ∨ x ∈ {Y}))
10	8, 9	sylib 188	. . . . . . 7 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (x ∈ A ∧ x ≠ Y)) → (x ∈ B ∨ x ∈ {Y}))
11		orel2 372	. . . . . . 7 ⊢ (¬ x ∈ {Y} → ((x ∈ B ∨ x ∈ {Y}) → x ∈ B))
12	3, 10, 11	sylc 56	. . . . . 6 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (x ∈ A ∧ x ≠ Y)) → x ∈ B)
13	12	ex 423	. . . . 5 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → ((x ∈ A ∧ x ≠ Y) → x ∈ B))
14		simp2l 981	. . . . . . . 8 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → ¬ X ∈ A)
15		eleq1 2413	. . . . . . . . 9 ⊢ (x = X → (x ∈ A ↔ X ∈ A))
16	15	notbid 285	. . . . . . . 8 ⊢ (x = X → (¬ x ∈ A ↔ ¬ X ∈ A))
17	14, 16	syl5ibrcom 213	. . . . . . 7 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → (x = X → ¬ x ∈ A))
18	17	necon2ad 2565	. . . . . 6 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → (x ∈ A → x ≠ X))
19	18	adantrd 454	. . . . 5 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → ((x ∈ A ∧ x ≠ Y) → x ≠ X))
20	13, 19	jcad 519	. . . 4 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → ((x ∈ A ∧ x ≠ Y) → (x ∈ B ∧ x ≠ X)))
21		eldifsn 3840	. . . 4 ⊢ (x ∈ (A ∖ {Y}) ↔ (x ∈ A ∧ x ≠ Y))
22		eldifsn 3840	. . . 4 ⊢ (x ∈ (B ∖ {X}) ↔ (x ∈ B ∧ x ≠ X))
23	20, 21, 22	3imtr4g 261	. . 3 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → (x ∈ (A ∖ {Y}) → x ∈ (B ∖ {X})))
24	23	ssrdv 3279	. 2 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → (A ∖ {Y}) ⊆ (B ∖ {X}))
25		elsni 3758	. . . . . . . . 9 ⊢ (x ∈ {X} → x = X)
26	25	necon3ai 2557	. . . . . . . 8 ⊢ (x ≠ X → ¬ x ∈ {X})
27	26	ad2antll 709	. . . . . . 7 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (x ∈ B ∧ x ≠ X)) → ¬ x ∈ {X})
28		ssun1 3427	. . . . . . . . . . 11 ⊢ B ⊆ (B ∪ {Y})
29	28	sseli 3270	. . . . . . . . . 10 ⊢ (x ∈ B → x ∈ (B ∪ {Y}))
30	29	ad2antrl 708	. . . . . . . . 9 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (x ∈ B ∧ x ≠ X)) → x ∈ (B ∪ {Y}))
31		simpl1 958	. . . . . . . . 9 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (x ∈ B ∧ x ≠ X)) → (A ∪ {X}) = (B ∪ {Y}))
32	30, 31	eleqtrrd 2430	. . . . . . . 8 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (x ∈ B ∧ x ≠ X)) → x ∈ (A ∪ {X}))
33		elun 3221	. . . . . . . 8 ⊢ (x ∈ (A ∪ {X}) ↔ (x ∈ A ∨ x ∈ {X}))
34	32, 33	sylib 188	. . . . . . 7 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (x ∈ B ∧ x ≠ X)) → (x ∈ A ∨ x ∈ {X}))
35		orel2 372	. . . . . . 7 ⊢ (¬ x ∈ {X} → ((x ∈ A ∨ x ∈ {X}) → x ∈ A))
36	27, 34, 35	sylc 56	. . . . . 6 ⊢ ((((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) ∧ (x ∈ B ∧ x ≠ X)) → x ∈ A)
37	36	ex 423	. . . . 5 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → ((x ∈ B ∧ x ≠ X) → x ∈ A))
38		simp2r 982	. . . . . . . 8 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → ¬ Y ∈ B)
39		eleq1 2413	. . . . . . . . 9 ⊢ (x = Y → (x ∈ B ↔ Y ∈ B))
40	39	notbid 285	. . . . . . . 8 ⊢ (x = Y → (¬ x ∈ B ↔ ¬ Y ∈ B))
41	38, 40	syl5ibrcom 213	. . . . . . 7 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → (x = Y → ¬ x ∈ B))
42	41	necon2ad 2565	. . . . . 6 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → (x ∈ B → x ≠ Y))
43	42	adantrd 454	. . . . 5 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → ((x ∈ B ∧ x ≠ X) → x ≠ Y))
44	37, 43	jcad 519	. . . 4 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → ((x ∈ B ∧ x ≠ X) → (x ∈ A ∧ x ≠ Y)))
45	44, 22, 21	3imtr4g 261	. . 3 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → (x ∈ (B ∖ {X}) → x ∈ (A ∖ {Y})))
46	45	ssrdv 3279	. 2 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → (B ∖ {X}) ⊆ (A ∖ {Y}))
47	24, 46	eqssd 3290	1 ⊢ (((A ∪ {X}) = (B ∪ {Y}) ∧ (¬ X ∈ A ∧ ¬ Y ∈ B) ∧ (Y ∈ A ∧ X ∈ B)) → (A ∖ {Y}) = (B ∖ {X}))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710   ≠ wne 2517   ∖ cdif 3207   ∪ cun 3208  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-sn 3742

This theorem is referenced by:  enadj  6061