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Theorem dftr6 25375
 Description: A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)
Hypothesis
Ref Expression
dftr6.1
Assertion
Ref Expression
dftr6

Proof of Theorem dftr6
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr6.1 . . . . 5
21elrn 5112 . . . 4
3 brdif 4262 . . . . . 6
4 vex 2961 . . . . . . . . 9
54, 1brco 5045 . . . . . . . 8
6 epel 4499 . . . . . . . . . 10
71epelc 4498 . . . . . . . . . 10
86, 7anbi12i 680 . . . . . . . . 9
98exbii 1593 . . . . . . . 8
105, 9bitri 242 . . . . . . 7
111epelc 4498 . . . . . . . 8
1211notbii 289 . . . . . . 7
1310, 12anbi12i 680 . . . . . 6
14 19.41v 1925 . . . . . . 7
15 exanali 1596 . . . . . . 7
1614, 15bitr3i 244 . . . . . 6
173, 13, 163bitri 264 . . . . 5
1817exbii 1593 . . . 4
19 exnal 1584 . . . 4
202, 18, 193bitri 264 . . 3
2120con2bii 324 . 2
22 dftr2 4306 . 2
23 eldif 3332 . . 3
241, 23mpbiran 886 . 2
2521, 22, 243bitr4i 270 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360  wal 1550  wex 1551   wcel 1726  cvv 2958   cdif 3319   class class class wbr 4214   wtr 4304   cep 4494   crn 4881   ccom 4884 This theorem is referenced by:  eltrans  25738 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891
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