Theorem erdisj2 10379

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 4267	. . . 4 |- (R = if(Er R, R, I) -> [A]R = [A]if(Er R, R, I))
2		eceq1 4267	. . . 4 |- (R = if(Er R, R, I) -> [B]R = [B]if(Er R, R, I))
3	1, 2	eqeq12d 1486	. . 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 2214	. . . 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 1480	. . 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 626	. 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 4257	. . . 4 |- (R = if(Er R, R, I) -> (Er R <-> Er if(Er R, R, I)))
10		ereq 4257	. . . 4 |- (I = if(Er R, R, I) -> (Er I <-> Er if(Er R, R, I)))
11		ider 4259	. . . 4 |- Er I
12	9, 10, 11	elimhyp 2386	. . 3 |- Er if(Er R, R, I)
13	7, 8, 12	erdisj 4276	. 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 2379	1 |- (Er R -> ([A]R = [B]R \/ ([A]R i^i [B]R) = (/)))

Syntax hints:   -> wi 3   \/ wo 222   = wceq 954   e. wcel 956  Vcvv 1807   i^i cin 2042  (/)c0 2276  ifcif 2357  Icid 2826  Er wer 4248  [cec 4249

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-er 4251  df-ec 4253

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