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Theorem issmo 6185
 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 5266 . . 3
41, 3mpbir 145 . 2
5 issmo.2 . . 3
6 ordeq 4294 . . . 4
72, 6ax-mp 5 . . 3
85, 7mpbir 145 . 2
92eleq2i 2206 . . . 4
102eleq2i 2206 . . . 4
11 issmo.3 . . . 4
129, 10, 11syl2anb 289 . . 3
1312rgen2a 2486 . 2
14 df-smo 6183 . 2
154, 8, 13, 14mpbir3an 1163 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1331   wcel 1480  wral 2416   word 4284  con0 4285   cdm 4539  wf 5119  cfv 5123   wsmo 6182 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288  df-fn 5126  df-f 5127  df-smo 6183 This theorem is referenced by:  iordsmo  6194
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