Theorem weinxp 3239

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 2066	. . . . . . . . . . . . . 14 ⊢ (z ⊆ A → (x ∈ z → x ∈ A))
2		ssel 2066	. . . . . . . . . . . . . 14 ⊢ (z ⊆ A → (y ∈ z → y ∈ A))
3	1, 2	anim12d 560	. . . . . . . . . . . . 13 ⊢ (z ⊆ A → ((x ∈ z ⋀ y ∈ z) → (x ∈ A ⋀ y ∈ A)))
4		brinxp 3238	. . . . . . . . . . . . . 14 ⊢ ((y ∈ A ⋀ x ∈ A) → (yRx ↔ y(R ∩ (A × A))x))
5	4	ancoms 438	. . . . . . . . . . . . 13 ⊢ ((x ∈ A ⋀ y ∈ A) → (yRx ↔ y(R ∩ (A × A))x))
6	3, 5	syl6 22	. . . . . . . . . . . 12 ⊢ (z ⊆ A → ((x ∈ z ⋀ y ∈ z) → (yRx ↔ y(R ∩ (A × A))x)))
7	6	exp3a 376	. . . . . . . . . . 11 ⊢ (z ⊆ A → (x ∈ z → (y ∈ z → (yRx ↔ y(R ∩ (A × A))x))))
8	7	imp31 362	. . . . . . . . . 10 ⊢ (((z ⊆ A ⋀ x ∈ z) ⋀ y ∈ z) → (yRx ↔ y(R ∩ (A × A))x))
9	8	negbid 613	. . . . . . . . 9 ⊢ (((z ⊆ A ⋀ x ∈ z) ⋀ y ∈ z) → (¬ yRx ↔ ¬ y(R ∩ (A × A))x))
10	9	ralbidva 1662	. . . . . . . 8 ⊢ ((z ⊆ A ⋀ x ∈ z) → (∀y ∈ z ¬ yRx ↔ ∀y ∈ z ¬ y(R ∩ (A × A))x))
11	10	rexbidva 1663	. . . . . . 7 ⊢ (z ⊆ A → (∃x ∈ z ∀y ∈ z ¬ yRx ↔ ∃x ∈ z ∀y ∈ z ¬ y(R ∩ (A × A))x))
12	11	adantr 391	. . . . . 6 ⊢ ((z ⊆ A ⋀ z ≠ ∅) → (∃x ∈ z ∀y ∈ z ¬ yRx ↔ ∃x ∈ z ∀y ∈ z ¬ y(R ∩ (A × A))x))
13	12	pm5.74i 586	. . . . 5 ⊢ (((z ⊆ A ⋀ z ≠ ∅) → ∃x ∈ z ∀y ∈ z ¬ yRx) ↔ ((z ⊆ A ⋀ z ≠ ∅) → ∃x ∈ z ∀y ∈ z ¬ y(R ∩ (A × A))x))
14	13	albii 1001	. . . 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 2923	. . . 4 ⊢ (R Fr A ↔ ∀z((z ⊆ A ⋀ z ≠ ∅) → ∃x ∈ z ∀y ∈ z ¬ yRx))
16		df-fr 2923	. . . 4 ⊢ ((R ∩ (A × A)) Fr A ↔ ∀z((z ⊆ A ⋀ z ≠ ∅) → ∃x ∈ z ∀y ∈ z ¬ y(R ∩ (A × A))x))
17	14, 15, 16	3bitr4 183	. . 3 ⊢ (R Fr A ↔ (R ∩ (A × A)) Fr A)
18		brinxp 3238	. . . . . . 7 ⊢ ((x ∈ A ⋀ y ∈ A) → (xRy ↔ x(R ∩ (A × A))y))
19		pm4.2d 171	. . . . . . 7 ⊢ ((x ∈ A ⋀ y ∈ A) → (x = y ↔ x = y))
20	18, 19, 5	3orbi123d 894	. . . . . 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 586	. . . . 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	2albii 1002	. . . 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 1679	. . . 4 ⊢ (∀x ∈ A ∀y ∈ A (xRy ⋁ x = y ⋁ yRx) ↔ ∀x∀y((x ∈ A ⋀ y ∈ A) → (xRy ⋁ x = y ⋁ yRx)))
24		r2al 1679	. . . 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 183	. . 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 484	. 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 2941	. 2 ⊢ (R We A ↔ (R Fr A ⋀ ∀x ∈ A ∀y ∈ A (xRy ⋁ x = y ⋁ yRx)))
28		dfwe2 2941	. 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 183	1 ⊢ (R We A ↔ (R ∩ (A × A)) We A)

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223   ⋁ w3o 776  ∀wal 956   = wceq 958   ∈ wcel 960   ≠ wne 1588  ∀wral 1648  ∃wrex 1649   ∩ cin 2049   ⊆ wss 2050  ∅c0 2283   class class class wbr 2624   Fr wfr 2921   We wwe 2922   × cxp 3174

This theorem is referenced by:  weth 4797

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-xp 3190

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