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Theorem erinxp 6180
 Description: A restricted equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 10-Jul-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
erinxp.r
erinxp.a
Assertion
Ref Expression
erinxp

Proof of Theorem erinxp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss2 3158 . . . 4
2 relxp 4447 . . . 4
3 relss 4427 . . . 4
41, 2, 3mp2 16 . . 3
54a1i 9 . 2
6 simpr 103 . . . . 5
7 brinxp2 4407 . . . . 5
86, 7sylib 127 . . . 4
98simp2d 917 . . 3
108simp1d 916 . . 3
11 erinxp.r . . . . 5
1211adantr 261 . . . 4
138simp3d 918 . . . 4
1412, 13ersym 6118 . . 3
15 brinxp2 4407 . . 3
169, 10, 14, 15syl3anbrc 1088 . 2
18 simprr 484 . . . . 5
19 brinxp2 4407 . . . . 5
2018, 19sylib 127 . . . 4
2120simp2d 917 . . 3
2211adantr 261 . . . 4
2313adantrr 448 . . . 4
2420simp3d 918 . . . 4
2522, 23, 24ertrd 6122 . . 3
26 brinxp2 4407 . . 3
2717, 21, 25, 26syl3anbrc 1088 . 2
2811adantr 261 . . . . . 6
29 erinxp.a . . . . . . 7
3029sselda 2945 . . . . . 6
3128, 30erref 6126 . . . . 5
3231ex 108 . . . 4
3332pm4.71rd 374 . . 3
34 brin 3811 . . . 4
35 brxp 4375 . . . . . 6
36 anidm 376 . . . . . 6
3735, 36bitri 173 . . . . 5
3837anbi2i 430 . . . 4
3934, 38bitri 173 . . 3
4033, 39syl6bbr 187 . 2
415, 16, 27, 40iserd 6132 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   w3a 885   wcel 1393   cin 2916   wss 2917   class class class wbr 3764   cxp 4343   wrel 4350   wer 6103 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-er 6106 This theorem is referenced by: (None)
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