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Theorem trpredeq2 25500
 Description: Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
Assertion
Ref Expression
trpredeq2

Proof of Theorem trpredeq2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 predeq2 25443 . . . . . . 7
21iuneq2d 4119 . . . . . 6
32mpteq2dv 4297 . . . . 5
4 predeq2 25443 . . . . 5
5 rdgeq12 6672 . . . . . 6
65reseq1d 5146 . . . . 5
73, 4, 6syl2anc 644 . . . 4
87rneqd 5098 . . 3
98unieqd 4027 . 2
10 df-trpred 25497 . 2
11 df-trpred 25497 . 2
129, 10, 113eqtr4g 2494 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653  cvv 2957  cuni 4016  ciun 4094   cmpt 4267  com 4846   crn 4880   cres 4881  crdg 6668  cpred 25439  ctrpred 25496 This theorem is referenced by:  trpredeq2d  25503 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-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418 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 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-xp 4885  df-cnv 4887  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fv 5463  df-recs 6634  df-rdg 6669  df-pred 25440  df-trpred 25497
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