Theorem erdisj2 10442

Description: Equivalence classes do not overlap.

Hypotheses

Ref	Expression
erdisj2.1	|- A e. V
erdisj2.2	|- B e. V

Assertion

Ref	Expression
erdisj2	|- (Er R -> ([A]R = [B]R \/ ([A]R i^i [B]R) = (/)))

Proof of Theorem erdisj2

Step	Hyp	Ref	Expression
1		eceq1 4277	. . . 4 |- (R = if(Er R, R, I) -> [A]R = [A]if(Er R, R, I))
2		eceq1 4277	. . . 4 |- (R = if(Er R, R, I) -> [B]R = [B]if(Er R, R, I))
3	1, 2	eqeq12d 1489	. . 3 |- (R = if(Er R, R, I) -> ([A]R = [B]R <-> [A]if(Er R, R, I) = [B]if(Er R, R, I)))
4	1, 2	ineq12d 2218	. . . 4 |- (R = if(Er R, R, I) -> ([A]R i^i [B]R) = ([A]if(Er R, R, I) i^i [B]if(Er R, R, I)))
5	4	eqeq1d 1483	. . 3 |- (R = if(Er R, R, I) -> (([A]R i^i [B]R) = (/) <-> ([A]if(Er R, R, I) i^i [B]if(Er R, R, I)) = (/)))
6	3, 5	orbi12d 627	. 2 |- (R = if(Er R, R, I) -> (([A]R = [B]R \/ ([A]R i^i [B]R) = (/)) <-> ([A]if(Er R, R, I) = [B]if(Er R, R, I) \/ ([A]if(Er R, R, I) i^i [B]if(Er R, R, I)) = (/))))
7		erdisj2.1	. . 3 |- A e. V
8		erdisj2.2	. . 3 |- B e. V
9		ereq 4267	. . . 4 |- (R = if(Er R, R, I) -> (Er R <-> Er if(Er R, R, I)))
10		ereq 4267	. . . 4 |- (I = if(Er R, R, I) -> (Er I <-> Er if(Er R, R, I)))
11		ider 4269	. . . 4 |- Er I
12	9, 10, 11	elimhyp 2390	. . 3 |- Er if(Er R, R, I)
13	7, 8, 12	erdisj 4286	. 2 |- ([A]if(Er R, R, I) = [B]if(Er R, R, I) \/ ([A]if(Er R, R, I) i^i [B]if(Er R, R, I)) = (/))
14	6, 13	dedth 2383	1 |- (Er R -> ([A]R = [B]R \/ ([A]R i^i [B]R) = (/)))

Syntax hints:   -> wi 3   \/ wo 222   = wceq 956   e. wcel 958  Vcvv 1811   i^i cin 2046  (/)c0 2280  ifcif 2361  Icid 2831  Er wer 4258  [cec 4259

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-er 4261  df-ec 4263

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