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Theorem issmo 7038
 Description: Conditions for which is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)
Hypotheses
Ref Expression
issmo.1
issmo.2
issmo.3
issmo.4
Assertion
Ref Expression
issmo
Distinct variable group:   ,,

Proof of Theorem issmo
StepHypRef Expression
1 issmo.1 . . 3
2 issmo.4 . . . 4
32feq2i 5729 . . 3
41, 3mpbir 209 . 2
5 issmo.2 . . 3
6 ordeq 4890 . . . 4
72, 6ax-mp 5 . . 3
85, 7mpbir 209 . 2
92eleq2i 2535 . . . 4
102eleq2i 2535 . . . 4
11 issmo.3 . . . 4
129, 10, 11syl2anb 479 . . 3
1312rgen2a 2884 . 2
14 df-smo 7036 . 2
154, 8, 13, 14mpbir3an 1178 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  Ordword 4882   con0 4883  domcdm 5004  -->wf 5589  cfv 5593  Smo`wsmo 7035 This theorem is referenced by:  iordsmo  7047  smobeth  8982 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-in 3482  df-ss 3489  df-uni 4250  df-tr 4546  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-fn 5596  df-f 5597  df-smo 7036
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