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Theorem prter1 32420
 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 4978 . . 3
32a1i 11 . 2
41prtlem16 32410 . . 3
54a1i 11 . 2
6 prtlem15 32416 . . . . . 6
71prtlem13 32409 . . . . . . . 8
81prtlem13 32409 . . . . . . . 8
97, 8anbi12i 701 . . . . . . 7
10 reeanv 2993 . . . . . . 7
119, 10bitr4i 255 . . . . . 6
121prtlem13 32409 . . . . . 6
136, 11, 123imtr4g 273 . . . . 5
14 pm3.22 450 . . . . . . 7
1514reximi 2890 . . . . . 6
161prtlem13 32409 . . . . . 6
1715, 7, 163imtr4i 269 . . . . 5
1813, 17jctil 539 . . . 4
1918alrimivv 1768 . . 3
2019alrimiv 1767 . 2
21 dfer2 7376 . 2
223, 5, 20, 21syl3anbrc 1189 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370  wal 1435   wceq 1437  wrex 2772  cuni 4219   class class class wbr 4423  copab 4481   cdm 4853   wrel 4858   wer 7372   wprt 32412 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-er 7375  df-prt 32413 This theorem is referenced by:  prtex  32421
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