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Theorem issmo 7020
 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 5724 . . 3
41, 3mpbir 209 . 2
5 issmo.2 . . 3
6 ordeq 4885 . . . 4
72, 6ax-mp 5 . . 3
85, 7mpbir 209 . 2
92eleq2i 2545 . . . 4
102eleq2i 2545 . . . 4
11 issmo.3 . . . 4
129, 10, 11syl2anb 479 . . 3
1312rgen2a 2891 . 2
14 df-smo 7018 . 2
154, 8, 13, 14mpbir3an 1178 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379   wcel 1767  wral 2814   word 4877  con0 4878   cdm 4999  wf 5584  cfv 5588   wsmo 7017 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-in 3483  df-ss 3490  df-uni 4246  df-tr 4541  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-fn 5591  df-f 5592  df-smo 7018 This theorem is referenced by:  iordsmo  7029  smobeth  8962
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