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Theorem prter1 26709
 Description: Every partition generates an equivalence relation. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypothesis
Ref Expression
prtlem18.1
Assertion
Ref Expression
prter1
Distinct variable group:   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem prter1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prtlem18.1 . . . 4
21relopabi 4992 . . 3
32a1i 11 . 2
41prtlem16 26699 . . 3
54a1i 11 . 2
6 prtlem15 26705 . . . . . 6
71prtlem13 26698 . . . . . . . 8
81prtlem13 26698 . . . . . . . 8
97, 8anbi12i 679 . . . . . . 7
10 reeanv 2867 . . . . . . 7
119, 10bitr4i 244 . . . . . 6
121prtlem13 26698 . . . . . 6
136, 11, 123imtr4g 262 . . . . 5
14 pm3.22 437 . . . . . . 7
1514reximi 2805 . . . . . 6
161prtlem13 26698 . . . . . 6
1715, 7, 163imtr4i 258 . . . . 5
1813, 17jctil 524 . . . 4
1918alrimivv 1642 . . 3
2019alrimiv 1641 . 2
21 dfer2 6898 . 2
223, 5, 20, 21syl3anbrc 1138 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1549   wceq 1652  wrex 2698  cuni 4007   class class class wbr 4204  copab 4257   cdm 4870   wrel 4875   wer 6894   wprt 26701 This theorem is referenced by:  prtex  26710 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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-er 6897  df-prt 26702
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