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Theorem trpredeq1 25437
 Description: Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
trpredeq1

Proof of Theorem trpredeq1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 predeq1 25383 . . . . . . . 8
21iuneq2d 4078 . . . . . . 7
32mpteq2dv 4256 . . . . . 6
4 predeq1 25383 . . . . . 6
5 rdgeq12 6630 . . . . . 6
63, 4, 5syl2anc 643 . . . . 5
76reseq1d 5104 . . . 4
87rneqd 5056 . . 3
98unieqd 3986 . 2
10 df-trpred 25435 . 2
11 df-trpred 25435 . 2
129, 10, 113eqtr4g 2461 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1649  cvv 2916  cuni 3975  ciun 4053   cmpt 4226  com 4804   crn 4838   cres 4839  crdg 6626  cpred 25381  ctrpred 25434 This theorem is referenced by:  trpredeq1d  25440 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fv 5421  df-recs 6592  df-rdg 6627  df-pred 25382  df-trpred 25435
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