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Theorem prter3 26731
 Description: For every partition there exists a unique equivalence relation whose quotient set equals the partition. (Contributed by Rodolfo Medina, 19-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
prtlem18.1
Assertion
Ref Expression
prter3
Distinct variable group:   ,,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem prter3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 errel 6914 . . 3
3 prtlem18.1 . . . 4
43relopabi 5000 . . 3
53prtlem13 26717 . . . . . 6
6 simpll 731 . . . . . . . . . . . . 13
7 simprl 733 . . . . . . . . . . . . . . 15
8 ne0i 3634 . . . . . . . . . . . . . . . 16
98ad2antll 710 . . . . . . . . . . . . . . 15
10 eldifsn 3927 . . . . . . . . . . . . . . 15
117, 9, 10sylanbrc 646 . . . . . . . . . . . . . 14
12 simplr 732 . . . . . . . . . . . . . 14
1311, 12eleqtrrd 2513 . . . . . . . . . . . . 13
14 simprr 734 . . . . . . . . . . . . 13
15 qsel 6983 . . . . . . . . . . . . 13
166, 13, 14, 15syl3anc 1184 . . . . . . . . . . . 12
1716eleq2d 2503 . . . . . . . . . . 11
18 vex 2959 . . . . . . . . . . . 12
19 vex 2959 . . . . . . . . . . . 12
2018, 19elec 6944 . . . . . . . . . . 11
2117, 20syl6bb 253 . . . . . . . . . 10
2221anassrs 630 . . . . . . . . 9
2322pm5.32da 623 . . . . . . . 8
2423rexbidva 2722 . . . . . . 7
25 simpll 731 . . . . . . . . . . . 12
26 simpr 448 . . . . . . . . . . . 12
2725, 26ercl 6916 . . . . . . . . . . 11
28 eluni2 4019 . . . . . . . . . . 11
2927, 28sylib 189 . . . . . . . . . 10
3029ex 424 . . . . . . . . 9
3130pm4.71rd 617 . . . . . . . 8
32 r19.41v 2861 . . . . . . . 8
3331, 32syl6bbr 255 . . . . . . 7
3424, 33bitr4d 248 . . . . . 6
355, 34syl5bb 249 . . . . 5
3635adantl 453 . . . 4
3736eqbrrdv2 26712 . . 3
384, 37mpanl1 662 . 2
392, 38mpancom 651 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725   wne 2599  wrex 2706   cdif 3317  c0 3628  csn 3814  cuni 4015   class class class wbr 4212  copab 4265   wrel 4883   wer 6902  cec 6903  cqs 6904 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-er 6905  df-ec 6907  df-qs 6911
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