Theorem erdisj 3223

Description: Equivalence classes do not overlap. In other words, two equivalence classes are either equal or disjoint. Theorem 74 of [Suppes] p. 83.

Hypotheses

Ref	Expression
erdisj.1	⊢ A ∈ V
erdisj.2	⊢ B ∈ V
erdisj.3	⊢ Er R

Assertion

Ref	Expression
erdisj	⊢ ([A]R = [B]R ∨ ([A]R ∩ [B]R) = ∅)

Proof of Theorem erdisj

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . 8 ⊢ x ∈ V
2		erdisj.1	. . . . . . . 8 ⊢ A ∈ V
3	1, 2	elec 3216	. . . . . . 7 ⊢ (x ∈ [A]R ↔ ARx)
4		erdisj.2	. . . . . . . . . . 11 ⊢ B ∈ V
5		erdisj.3	. . . . . . . . . . 11 ⊢ Er R
6	2, 1, 4, 5	ertr 3211	. . . . . . . . . 10 ⊢ ((ARx ∧ xRB) → ARB)
7	6	exp 291	. . . . . . . . 9 ⊢ (ARx → (xRB → ARB))
8	2, 4, 5	erthi 3218	. . . . . . . . 9 ⊢ (ARB → [A]R = [B]R)
9	7, 8	syl6 23	. . . . . . . 8 ⊢ (ARx → (xRB → [A]R = [B]R))
10	1, 4	elec 3216	. . . . . . . . 9 ⊢ (x ∈ [B]R ↔ BRx)
11	4, 1, 5	ersymb 3210	. . . . . . . . 9 ⊢ (BRx ↔ xRB)
12	10, 11	bitr 151	. . . . . . . 8 ⊢ (x ∈ [B]R ↔ xRB)
13	9, 12	syl5ib 181	. . . . . . 7 ⊢ (ARx → (x ∈ [B]R → [A]R = [B]R))
14	3, 13	sylbi 174	. . . . . 6 ⊢ (x ∈ [A]R → (x ∈ [B]R → [A]R = [B]R))
15	14	con3d 87	. . . . 5 ⊢ (x ∈ [A]R → (¬ [A]R = [B]R → ¬ x ∈ [B]R))
16	15	com12 13	. . . 4 ⊢ (¬ [A]R = [B]R → (x ∈ [A]R → ¬ x ∈ [B]R))
17	16	19.21aiv 943	. . 3 ⊢ (¬ [A]R = [B]R → ∀x(x ∈ [A]R → ¬ x ∈ [B]R))
18		disj1 1734	. . 3 ⊢ (([A]R ∩ [B]R) = ∅ ↔ ∀x(x ∈ [A]R → ¬ x ∈ [B]R))
19	17, 18	sylibr 175	. 2 ⊢ (¬ [A]R = [B]R → ([A]R ∩ [B]R) = ∅)
20	19	orri 201	1 ⊢ ([A]R = [B]R ∨ ([A]R ∩ [B]R) = ∅)

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195  ∀wal 672   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∩ cin 1486  ∅c0 1707   class class class wbr 2054  Er wer 3197  [cec 3198

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  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-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-er 3200  df-ec 3202

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