Theorem erdisj 5344

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 2295	. . . . . . . 8 |- x e. _V
2		erdisj.1	. . . . . . . 8 |- A e. _V
3	1, 2	elec 5337	. . . . . . 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 5332	. . . . . . . . . 10 |- ((ARx /\ xRB) -> ARB)
7	6	ex 402	. . . . . . . . 9 |- (ARx -> (xRB -> ARB))
8	2, 4, 5	erthi 5339	. . . . . . . . 9 |- (ARB -> [A]R = [B]R)
9	7, 8	syl6 25	. . . . . . . 8 |- (ARx -> (xRB -> [A]R = [B]R))
10	1, 4	elec 5337	. . . . . . . . 9 |- (x e. [B]R <-> BRx)
11	4, 1, 5	ersymb 5331	. . . . . . . . 9 |- (BRx <-> xRB)
12	10, 11	bitri 190	. . . . . . . 8 |- (x e. [B]R <-> xRB)
13	9, 12	syl5ib 223	. . . . . . 7 |- (ARx -> (x e. [B]R -> [A]R = [B]R))
14	3, 13	sylbi 216	. . . . . 6 |- (x e. [A]R -> (x e. [B]R -> [A]R = [B]R))
15	14	con3d 111	. . . . 5 |- (x e. [A]R -> (-. [A]R = [B]R -> -. x e. [B]R))
16	15	com12 14	. . . 4 |- (-. [A]R = [B]R -> (x e. [A]R -> -. x e. [B]R))
17	16	19.21aiv 1664	. . 3 |- (-. [A]R = [B]R -> A.x(x e. [A]R -> -. x e. [B]R))
18		disj1 2915	. . 3 |- (([A]R i^i [B]R) = (/) <-> A.x(x e. [A]R -> -. x e. [B]R))
19	17, 18	sylibr 217	. 2 |- (-. [A]R = [B]R -> ([A]R i^i [B]R) = (/))
20	19	orri 248	1 |- ([A]R = [B]R \/ ([A]R i^i [B]R) = (/))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239  A.wal 1296   = wceq 1298   e. wcel 1300  _Vcvv 2292   i^i cin 2592  (/)c0 2875   class class class wbr 3338  Er wer 5315  [cec 5316

This theorem is referenced by:  erdisj2 10164  uninqs 14340

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-er 5318  df-ec 5320

Copyright terms: Public domain