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Theorem trpredeq1 25503
 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 25446 . . . . . . . 8
21iuneq2d 4120 . . . . . . 7
32mpteq2dv 4299 . . . . . 6
4 predeq1 25446 . . . . . 6
5 rdgeq12 6674 . . . . . 6
63, 4, 5syl2anc 644 . . . . 5
76reseq1d 5148 . . . 4
87rneqd 5100 . . 3
98unieqd 4028 . 2
10 df-trpred 25501 . 2
11 df-trpred 25501 . 2
129, 10, 113eqtr4g 2495 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653  cvv 2958  cuni 4017  ciun 4095   cmpt 4269  com 4848   crn 4882   cres 4883  crdg 6670  cpred 25443  ctrpred 25500 This theorem is referenced by:  trpredeq1d  25506 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 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-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  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-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-xp 4887  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fv 5465  df-recs 6636  df-rdg 6671  df-pred 25444  df-trpred 25501
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