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

Theorem issmo 6365
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 5384 . . 3  |-  ( A : dom  A --> On  <->  A : B
--> On )
41, 3mpbir 200 . 2  |-  A : dom  A --> On
5 issmo.2 . . 3  |-  Ord  B
6 ordeq 4399 . . . 4  |-  ( dom 
A  =  B  -> 
( Ord  dom  A  <->  Ord  B ) )
72, 6ax-mp 8 . . 3  |-  ( Ord 
dom  A  <->  Ord  B )
85, 7mpbir 200 . 2  |-  Ord  dom  A
92eleq2i 2347 . . . 4  |-  ( x  e.  dom  A  <->  x  e.  B )
102eleq2i 2347 . . . 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 465 . . 3  |-  ( ( x  e.  dom  A  /\  y  e.  dom  A )  ->  ( x  e.  y  ->  ( A `
 x )  e.  ( A `  y
) ) )
1312rgen2a 2609 . 2  |-  A. x  e.  dom  A A. y  e.  dom  A ( x  e.  y  ->  ( A `  x )  e.  ( A `  y
) )
14 df-smo 6363 . 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 1134 1  |-  Smo  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   Ord word 4391   Oncon0 4392   dom cdm 4689   -->wf 5251   ` cfv 5255   Smo wsmo 6362
This theorem is referenced by:  iordsmo  6374  smobeth  8208
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-in 3159  df-ss 3166  df-uni 3828  df-tr 4114  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-fn 5258  df-f 5259  df-smo 6363
  Copyright terms: Public domain W3C validator