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

Theorem issmo 7075
Description: Conditions for which  A is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)
Hypotheses
Ref Expression
issmo.1  |-  A : B
--> On
issmo.2  |-  Ord  B
issmo.3  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  e.  y  ->  ( A `  x )  e.  ( A `  y ) ) )
issmo.4  |-  dom  A  =  B
Assertion
Ref Expression
issmo  |-  Smo  A
Distinct variable group:    x, y, A
Allowed substitution hints:    B( x, y)

Proof of Theorem issmo
StepHypRef Expression
1 issmo.1 . . 3  |-  A : B
--> On
2 issmo.4 . . . 4  |-  dom  A  =  B
32feq2i 5739 . . 3  |-  ( A : dom  A --> On  <->  A : B
--> On )
41, 3mpbir 212 . 2  |-  A : dom  A --> On
5 issmo.2 . . 3  |-  Ord  B
6 ordeq 5449 . . . 4  |-  ( dom 
A  =  B  -> 
( Ord  dom  A  <->  Ord  B ) )
72, 6ax-mp 5 . . 3  |-  ( Ord 
dom  A  <->  Ord  B )
85, 7mpbir 212 . 2  |-  Ord  dom  A
92eleq2i 2507 . . . 4  |-  ( x  e.  dom  A  <->  x  e.  B )
102eleq2i 2507 . . . 4  |-  ( y  e.  dom  A  <->  y  e.  B )
11 issmo.3 . . . 4  |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  e.  y  ->  ( A `  x )  e.  ( A `  y ) ) )
129, 10, 11syl2anb 481 . . 3  |-  ( ( x  e.  dom  A  /\  y  e.  dom  A )  ->  ( x  e.  y  ->  ( A `
 x )  e.  ( A `  y
) ) )
1312rgen2a 2859 . 2  |-  A. x  e.  dom  A A. y  e.  dom  A ( x  e.  y  ->  ( A `  x )  e.  ( A `  y
) )
14 df-smo 7073 . 2  |-  ( Smo 
A  <->  ( A : dom  A --> On  /\  Ord  dom 
A  /\  A. x  e.  dom  A A. y  e.  dom  A ( x  e.  y  ->  ( A `  x )  e.  ( A `  y
) ) ) )
154, 8, 13, 14mpbir3an 1187 1  |-  Smo  A
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   dom cdm 4854   Ord word 5441   Oncon0 5442   -->wf 5597   ` cfv 5601   Smo 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
  Copyright terms: Public domain W3C validator