Theorem wereu 2197

Description: A subset of a well-ordered set has a unique minimal element.

Hypothesis

Ref	Expression
wereu.1	⊢ B ∈ V

Assertion

Ref	Expression
wereu	⊢ ((R We A ∧ (B ⊆ A ∧ ¬ B = ∅)) → ∃!x ∈ B ∀y ∈ B ¬ yRx)

Distinct variable group(s):   x,y,R   x,A,y   x,B,y

Proof of Theorem wereu

Step	Hyp	Ref	Expression
1		wereu.1	. . . . 5 ⊢ B ∈ V
2	1	fri 2170	. . . 4 ⊢ ((R Fr A ∧ (B ⊆ A ∧ ¬ B = ∅)) → ∃x ∈ B ∀y ∈ B ¬ yRx)
3		wefr 2191	. . . 4 ⊢ (R We A → R Fr A)
4	2, 3	sylan 343	. . 3 ⊢ ((R We A ∧ (B ⊆ A ∧ ¬ B = ∅)) → ∃x ∈ B ∀y ∈ B ¬ yRx)
5		solin 2145	. . . . . . . . . . . . 13 ⊢ ((R Or B ∧ (x ∈ B ∧ z ∈ B)) → (xRz ∨ x = z ∨ zRx))
6		weso 2192	. . . . . . . . . . . . 13 ⊢ (R We B → R Or B)
7	5, 6	sylan 343	. . . . . . . . . . . 12 ⊢ ((R We B ∧ (x ∈ B ∧ z ∈ B)) → (xRz ∨ x = z ∨ zRx))
8		df-3or 582	. . . . . . . . . . . . 13 ⊢ ((xRz ∨ x = z ∨ zRx) ↔ ((xRz ∨ x = z) ∨ zRx))
9		or23 219	. . . . . . . . . . . . 13 ⊢ (((xRz ∨ x = z) ∨ zRx) ↔ ((xRz ∨ zRx) ∨ x = z))
10		df-or 197	. . . . . . . . . . . . 13 ⊢ (((xRz ∨ zRx) ∨ x = z) ↔ (¬ (xRz ∨ zRx) → x = z))
11	8, 9, 10	3bitr 155	. . . . . . . . . . . 12 ⊢ ((xRz ∨ x = z ∨ zRx) ↔ (¬ (xRz ∨ zRx) → x = z))
12	7, 11	sylib 173	. . . . . . . . . . 11 ⊢ ((R We B ∧ (x ∈ B ∧ z ∈ B)) → (¬ (xRz ∨ zRx) → x = z))
13		ioran 254	. . . . . . . . . . 11 ⊢ (¬ (xRz ∨ zRx) ↔ (¬ xRz ∧ ¬ zRx))
14	12, 13	syl5ibr 182	. . . . . . . . . 10 ⊢ ((R We B ∧ (x ∈ B ∧ z ∈ B)) → ((¬ xRz ∧ ¬ zRx) → x = z))
15		wess 2188	. . . . . . . . . . . 12 ⊢ (B ⊆ A → (R We A → R We B))
16	15	imp 277	. . . . . . . . . . 11 ⊢ ((B ⊆ A ∧ R We A) → R We B)
17	16	ancoms 334	. . . . . . . . . 10 ⊢ ((R We A ∧ B ⊆ A) → R We B)
18	14, 17	sylan 343	. . . . . . . . 9 ⊢ (((R We A ∧ B ⊆ A) ∧ (x ∈ B ∧ z ∈ B)) → ((¬ xRz ∧ ¬ zRx) → x = z))
19		breq1 2065	. . . . . . . . . . . . . 14 ⊢ (y = x → (yRz ↔ xRz))
20	19	negbid 463	. . . . . . . . . . . . 13 ⊢ (y = x → (¬ yRz ↔ ¬ xRz))
21	20	rcla4v 1402	. . . . . . . . . . . 12 ⊢ (∀y ∈ B ¬ yRz → (x ∈ B → ¬ xRz))
22		breq1 2065	. . . . . . . . . . . . . 14 ⊢ (y = z → (yRx ↔ zRx))
23	22	negbid 463	. . . . . . . . . . . . 13 ⊢ (y = z → (¬ yRx ↔ ¬ zRx))
24	23	rcla4v 1402	. . . . . . . . . . . 12 ⊢ (∀y ∈ B ¬ yRx → (z ∈ B → ¬ zRx))
25	21, 24	im2anan9r 435	. . . . . . . . . . 11 ⊢ ((∀y ∈ B ¬ yRx ∧ ∀y ∈ B ¬ yRz) → ((x ∈ B ∧ z ∈ B) → (¬ xRz ∧ ¬ zRx)))
26	25	imp 277	. . . . . . . . . 10 ⊢ (((∀y ∈ B ¬ yRx ∧ ∀y ∈ B ¬ yRz) ∧ (x ∈ B ∧ z ∈ B)) → (¬ xRz ∧ ¬ zRx))
27	26	ancoms 334	. . . . . . . . 9 ⊢ (((x ∈ B ∧ z ∈ B) ∧ (∀y ∈ B ¬ yRx ∧ ∀y ∈ B ¬ yRz)) → (¬ xRz ∧ ¬ zRx))
28	18, 27	syl5 22	. . . . . . . 8 ⊢ (((R We A ∧ B ⊆ A) ∧ (x ∈ B ∧ z ∈ B)) → (((x ∈ B ∧ z ∈ B) ∧ (∀y ∈ B ¬ yRx ∧ ∀y ∈ B ¬ yRz)) → x = z))
29	28	exp4b 296	. . . . . . 7 ⊢ ((R We A ∧ B ⊆ A) → ((x ∈ B ∧ z ∈ B) → ((x ∈ B ∧ z ∈ B) → ((∀y ∈ B ¬ yRx ∧ ∀y ∈ B ¬ yRz) → x = z))))
30	29	pm2.43d 59	. . . . . 6 ⊢ ((R We A ∧ B ⊆ A) → ((x ∈ B ∧ z ∈ B) → ((∀y ∈ B ¬ yRx ∧ ∀y ∈ B ¬ yRz) → x = z)))
31	30	adantrr 312	. . . . 5 ⊢ ((R We A ∧ (B ⊆ A ∧ ¬ B = ∅)) → ((x ∈ B ∧ z ∈ B) → ((∀y ∈ B ¬ yRx ∧ ∀y ∈ B ¬ yRz) → x = z)))
32	31	19.21aivv 944	. . . 4 ⊢ ((R We A ∧ (B ⊆ A ∧ ¬ B = ∅)) → ∀x∀z((x ∈ B ∧ z ∈ B) → ((∀y ∈ B ¬ yRx ∧ ∀y ∈ B ¬ yRz) → x = z)))
33		r2al 1231	. . . 4 ⊢ (∀x ∈ B ∀z ∈ B ((∀y ∈ B ¬ yRx ∧ ∀y ∈ B ¬ yRz) → x = z) ↔ ∀x∀z((x ∈ B ∧ z ∈ B) → ((∀y ∈ B ¬ yRx ∧ ∀y ∈ B ¬ yRz) → x = z)))
34	32, 33	sylibr 175	. . 3 ⊢ ((R We A ∧ (B ⊆ A ∧ ¬ B = ∅)) → ∀x ∈ B ∀z ∈ B ((∀y ∈ B ¬ yRx ∧ ∀y ∈ B ¬ yRz) → x = z))
35	4, 34	jca 236	. 2 ⊢ ((R We A ∧ (B ⊆ A ∧ ¬ B = ∅)) → (∃x ∈ B ∀y ∈ B ¬ yRx ∧ ∀x ∈ B ∀z ∈ B ((∀y ∈ B ¬ yRx ∧ ∀y ∈ B ¬ yRz) → x = z)))
36		breq2 2066	. . . . 5 ⊢ (x = z → (yRx ↔ yRz))
37	36	negbid 463	. . . 4 ⊢ (x = z → (¬ yRx ↔ ¬ yRz))
38	37	biraldv 1219	. . 3 ⊢ (x = z → (∀y ∈ B ¬ yRx ↔ ∀y ∈ B ¬ yRz))
39	38	reu4 1340	. 2 ⊢ (∃!x ∈ B ∀y ∈ B ¬ yRx ↔ (∃x ∈ B ∀y ∈ B ¬ yRx ∧ ∀x ∈ B ∀z ∈ B ((∀y ∈ B ¬ yRx ∧ ∀y ∈ B ¬ yRz) → x = z)))
40	35, 39	sylibr 175	1 ⊢ ((R We A ∧ (B ⊆ A ∧ ¬ B = ∅)) → ∃!x ∈ B ∀y ∈ B ¬ yRx)

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196   ∨ w3o 580  ∀wal 672   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  ∃!wreu 1203  Vcvv 1348   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054   Or wor 2059   Fr wfr 2061   We wwe 2062

This theorem is referenced by:  zornlem1 3603  htalem 3618

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-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-po 2128  df-so 2138  df-fr 2169  df-we 2186

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