Theorem weinxp 2467

Description: Intersection of well-ordering with cross product of its field.

Assertion

Ref	Expression
weinxp	⊢ (R We A ↔ (R ∩ (A × A)) We A)

Proof of Theorem weinxp

Step	Hyp	Ref	Expression
1		ssel 1502	. . . . . . . . . . . . . 14 ⊢ (z ⊆ A → (x ∈ z → x ∈ A))
2		ssel 1502	. . . . . . . . . . . . . 14 ⊢ (z ⊆ A → (y ∈ z → y ∈ A))
3	1, 2	anim12d 431	. . . . . . . . . . . . 13 ⊢ (z ⊆ A → ((x ∈ z ∧ y ∈ z) → (x ∈ A ∧ y ∈ A)))
4	3	adantr 306	. . . . . . . . . . . 12 ⊢ ((z ⊆ A ∧ ¬ z = ∅) → ((x ∈ z ∧ y ∈ z) → (x ∈ A ∧ y ∈ A)))
5		brinxp 2466	. . . . . . . . . . . . 13 ⊢ ((y ∈ A ∧ x ∈ A) → (yRx ↔ y(R ∩ (A × A))x))
6	5	ancoms 334	. . . . . . . . . . . 12 ⊢ ((x ∈ A ∧ y ∈ A) → (yRx ↔ y(R ∩ (A × A))x))
7	4, 6	syl6 23	. . . . . . . . . . 11 ⊢ ((z ⊆ A ∧ ¬ z = ∅) → ((x ∈ z ∧ y ∈ z) → (yRx ↔ y(R ∩ (A × A))x)))
8	7	exp3a 292	. . . . . . . . . 10 ⊢ ((z ⊆ A ∧ ¬ z = ∅) → (x ∈ z → (y ∈ z → (yRx ↔ y(R ∩ (A × A))x))))
9	8	imp31 280	. . . . . . . . 9 ⊢ ((((z ⊆ A ∧ ¬ z = ∅) ∧ x ∈ z) ∧ y ∈ z) → (yRx ↔ y(R ∩ (A × A))x))
10	9	negbid 463	. . . . . . . 8 ⊢ ((((z ⊆ A ∧ ¬ z = ∅) ∧ x ∈ z) ∧ y ∈ z) → (¬ yRx ↔ ¬ y(R ∩ (A × A))x))
11	10	biraldva 1215	. . . . . . 7 ⊢ (((z ⊆ A ∧ ¬ z = ∅) ∧ x ∈ z) → (∀y ∈ z ¬ yRx ↔ ∀y ∈ z ¬ y(R ∩ (A × A))x))
12	11	birexdva 1216	. . . . . 6 ⊢ ((z ⊆ A ∧ ¬ z = ∅) → (∃x ∈ z ∀y ∈ z ¬ yRx ↔ ∃x ∈ z ∀y ∈ z ¬ y(R ∩ (A × A))x))
13	12	pm5.74i 443	. . . . 5 ⊢ (((z ⊆ A ∧ ¬ z = ∅) → ∃x ∈ z ∀y ∈ z ¬ yRx) ↔ ((z ⊆ A ∧ ¬ z = ∅) → ∃x ∈ z ∀y ∈ z ¬ y(R ∩ (A × A))x))
14	13	bial 695	. . . 4 ⊢ (∀z((z ⊆ A ∧ ¬ z = ∅) → ∃x ∈ z ∀y ∈ z ¬ yRx) ↔ ∀z((z ⊆ A ∧ ¬ z = ∅) → ∃x ∈ z ∀y ∈ z ¬ y(R ∩ (A × A))x))
15		df-fr 2169	. . . 4 ⊢ (R Fr A ↔ ∀z((z ⊆ A ∧ ¬ z = ∅) → ∃x ∈ z ∀y ∈ z ¬ yRx))
16		df-fr 2169	. . . 4 ⊢ ((R ∩ (A × A)) Fr A ↔ ∀z((z ⊆ A ∧ ¬ z = ∅) → ∃x ∈ z ∀y ∈ z ¬ y(R ∩ (A × A))x))
17	14, 15, 16	3bitr4 158	. . 3 ⊢ (R Fr A ↔ (R ∩ (A × A)) Fr A)
18		brinxp 2466	. . . . . . 7 ⊢ ((x ∈ A ∧ y ∈ A) → (xRy ↔ x(R ∩ (A × A))y))
19		pm4.2i 149	. . . . . . 7 ⊢ ((x ∈ A ∧ y ∈ A) → (x = y ↔ x = y))
20	18, 19, 6	bi3ord 635	. . . . . 6 ⊢ ((x ∈ A ∧ y ∈ A) → ((xRy ∨ x = y ∨ yRx) ↔ (x(R ∩ (A × A))y ∨ x = y ∨ y(R ∩ (A × A))x)))
21	20	pm5.74i 443	. . . . 5 ⊢ (((x ∈ A ∧ y ∈ A) → (xRy ∨ x = y ∨ yRx)) ↔ ((x ∈ A ∧ y ∈ A) → (x(R ∩ (A × A))y ∨ x = y ∨ y(R ∩ (A × A))x)))
22	21	bi2al 696	. . . 4 ⊢ (∀x∀y((x ∈ A ∧ y ∈ A) → (xRy ∨ x = y ∨ yRx)) ↔ ∀x∀y((x ∈ A ∧ y ∈ A) → (x(R ∩ (A × A))y ∨ x = y ∨ y(R ∩ (A × A))x)))
23		r2al 1231	. . . 4 ⊢ (∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx) ↔ ∀x∀y((x ∈ A ∧ y ∈ A) → (xRy ∨ x = y ∨ yRx)))
24		r2al 1231	. . . 4 ⊢ (∀x ∈ A ∀y ∈ A (x(R ∩ (A × A))y ∨ x = y ∨ y(R ∩ (A × A))x) ↔ ∀x∀y((x ∈ A ∧ y ∈ A) → (x(R ∩ (A × A))y ∨ x = y ∨ y(R ∩ (A × A))x)))
25	22, 23, 24	3bitr4 158	. . 3 ⊢ (∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx) ↔ ∀x ∈ A ∀y ∈ A (x(R ∩ (A × A))y ∨ x = y ∨ y(R ∩ (A × A))x))
26	17, 25	anbi12i 369	. 2 ⊢ ((R Fr A ∧ ∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx)) ↔ ((R ∩ (A × A)) Fr A ∧ ∀x ∈ A ∀y ∈ A (x(R ∩ (A × A))y ∨ x = y ∨ y(R ∩ (A × A))x)))
27		dfwe2 2187	. 2 ⊢ (R We A ↔ (R Fr A ∧ ∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx)))
28		dfwe2 2187	. 2 ⊢ ((R ∩ (A × A)) We A ↔ ((R ∩ (A × A)) Fr A ∧ ∀x ∈ A ∀y ∈ A (x(R ∩ (A × A))y ∨ x = y ∨ y(R ∩ (A × A))x)))
29	26, 27, 28	3bitr4 158	1 ⊢ (R We A ↔ (R ∩ (A × A)) We A)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   ∨ w3o 580  ∀wal 672   = weq 797   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054   Fr wfr 2061   We wwe 2062   × cxp 2408

This theorem is referenced by:  weth 3602

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

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-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-xp 2424

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