Theorem erdisj 4292

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 1816	. . . . . . . 8 |- x e. V
2		erdisj.1	. . . . . . . 8 |- A e. V
3	1, 2	elec 4285	. . . . . . 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 4280	. . . . . . . . . 10 |- ((ARx /\ xRB) -> ARB)
7	6	ex 373	. . . . . . . . 9 |- (ARx -> (xRB -> ARB))
8	2, 4, 5	erthi 4287	. . . . . . . . 9 |- (ARB -> [A]R = [B]R)
9	7, 8	syl6 22	. . . . . . . 8 |- (ARx -> (xRB -> [A]R = [B]R))
10	1, 4	elec 4285	. . . . . . . . 9 |- (x e. [B]R <-> BRx)
11	4, 1, 5	ersymb 4279	. . . . . . . . 9 |- (BRx <-> xRB)
12	10, 11	bitr 173	. . . . . . . 8 |- (x e. [B]R <-> xRB)
13	9, 12	syl5ib 206	. . . . . . 7 |- (ARx -> (x e. [B]R -> [A]R = [B]R))
14	3, 13	sylbi 199	. . . . . 6 |- (x e. [A]R -> (x e. [B]R -> [A]R = [B]R))
15	14	con3d 95	. . . . 5 |- (x e. [A]R -> (-. [A]R = [B]R -> -. x e. [B]R))
16	15	com12 11	. . . 4 |- (-. [A]R = [B]R -> (x e. [A]R -> -. x e. [B]R))
17	16	19.21aiv 1288	. . 3 |- (-. [A]R = [B]R -> A.x(x e. [A]R -> -. x e. [B]R))
18		disj1 2316	. . 3 |- (([A]R i^i [B]R) = (/) <-> A.x(x e. [A]R -> -. x e. [B]R))
19	17, 18	sylibr 200	. 2 |- (-. [A]R = [B]R -> ([A]R i^i [B]R) = (/))
20	19	orri 231	1 |- ([A]R = [B]R \/ ([A]R i^i [B]R) = (/))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222  A.wal 956   = wceq 958   e. wcel 960  Vcvv 1814   i^i cin 2049  (/)c0 2283   class class class wbr 2624  Er wer 4264  [cec 4265

This theorem is referenced by:  uninqs 10436  erdisj2 10437

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-9 967  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-er 4267  df-ec 4269

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