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Theorem issmo 7075
 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:   ,,
Allowed substitution hints:   (,)

Proof of Theorem issmo
StepHypRef Expression
1 issmo.1 . . 3
2 issmo.4 . . . 4
32feq2i 5739 . . 3
41, 3mpbir 212 . 2
5 issmo.2 . . 3
6 ordeq 5449 . . . 4
72, 6ax-mp 5 . . 3
85, 7mpbir 212 . 2
92eleq2i 2507 . . . 4
102eleq2i 2507 . . . 4
11 issmo.3 . . . 4
129, 10, 11syl2anb 481 . . 3
1312rgen2a 2859 . 2
14 df-smo 7073 . 2
154, 8, 13, 14mpbir3an 1187 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437   wcel 1870  wral 2782   cdm 4854   word 5441  con0 5442  wf 5597  cfv 5601   wsmo 7072 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407 This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-in 3449  df-ss 3456  df-uni 4223  df-tr 4521  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-ord 5445  df-fn 5604  df-f 5605  df-smo 7073 This theorem is referenced by:  iordsmo  7084  smobeth  9009
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