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 e. V
erdisj.2	|- B e. V
erdisj.3	|- Er R

Assertion

Ref	Expression
erdisj	|- ([A]R = [B]R \/ ([A]R i^i [B]R) = (/))

Proof of Theorem erdisj

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . 8 |- x e. V
2		erdisj.1	. . . . . . . 8 |- A e. V
3	1, 2	elec 3216	. . . . . . 7 |- (x e. [A]R <-> ARx)
4		erdisj.2	. . . . . . . . . . 11 |- B e. 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 e. [B]R <-> BRx)
11	4, 1, 5	ersymb 3210	. . . . . . . . 9 |- (BRx <-> xRB)
12	10, 11	bitr 151	. . . . . . . 8 |- (x e. [B]R <-> xRB)
13	9, 12	syl5ib 181	. . . . . . 7 |- (ARx -> (x e. [B]R -> [A]R = [B]R))
14	3, 13	sylbi 174	. . . . . 6 |- (x e. [A]R -> (x e. [B]R -> [A]R = [B]R))
15	14	con3d 87	. . . . 5 |- (x e. [A]R -> (-. [A]R = [B]R -> -. x e. [B]R))
16	15	com12 13	. . . 4 |- (-. [A]R = [B]R -> (x e. [A]R -> -. x e. [B]R))
17	16	19.21aiv 943	. . 3 |- (-. [A]R = [B]R -> A.x(x e. [A]R -> -. x e. [B]R))
18		disj1 1734	. . 3 |- (([A]R i^i [B]R) = (/) <-> A.x(x e. [A]R -> -. x e. [B]R))
19	17, 18	sylibr 175	. 2 |- (-. [A]R = [B]R -> ([A]R i^i [B]R) = (/))
20	19	orri 201	1 |- ([A]R = [B]R \/ ([A]R i^i [B]R) = (/))

Syntax hints:  -. wn 1   -> wi 2   \/ wo 195  A.wal 672   = wceq 1091   e. wcel 1092  Vcvv 1348   i^i 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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