Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  issmo Structured version   Unicode version

Theorem issmo 6602
 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 5578 . . 3
41, 3mpbir 201 . 2
5 issmo.2 . . 3
6 ordeq 4580 . . . 4
72, 6ax-mp 8 . . 3
85, 7mpbir 201 . 2
92eleq2i 2499 . . . 4
102eleq2i 2499 . . . 4
11 issmo.3 . . . 4
129, 10, 11syl2anb 466 . . 3
1312rgen2a 2764 . 2
14 df-smo 6600 . 2
154, 8, 13, 14mpbir3an 1136 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2697   word 4572  con0 4573   cdm 4870  wf 5442  cfv 5446   wsmo 6599 This theorem is referenced by:  iordsmo  6611  smobeth  8453 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-in 3319  df-ss 3326  df-uni 4008  df-tr 4295  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-fn 5449  df-f 5450  df-smo 6600
 Copyright terms: Public domain W3C validator