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Theorem ordtriexmidlem 4530
Description: Lemma for decidability and ordinals. The set  { x  e.  { (/)
}  |  ph } is a way of connecting statements about ordinals (such as trichotomy in ordtriexmid 4532 or weak linearity in ordsoexmid 4573) with a proposition  ph. Our lemma states that it is an ordinal number. (Contributed by Jim Kingdon, 28-Jan-2019.)
Assertion
Ref Expression
ordtriexmidlem  |-  { x  e.  { (/) }  |  ph }  e.  On

Proof of Theorem ordtriexmidlem
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 109 . . . . . 6  |-  ( ( y  e.  z  /\  z  e.  { x  e.  { (/) }  |  ph } )  ->  y  e.  z )
2 elrabi 2902 . . . . . . . . 9  |-  ( z  e.  { x  e. 
{ (/) }  |  ph }  ->  z  e.  { (/)
} )
3 velsn 3621 . . . . . . . . 9  |-  ( z  e.  { (/) }  <->  z  =  (/) )
42, 3sylib 122 . . . . . . . 8  |-  ( z  e.  { x  e. 
{ (/) }  |  ph }  ->  z  =  (/) )
5 noel 3438 . . . . . . . . 9  |-  -.  y  e.  (/)
6 eleq2 2251 . . . . . . . . 9  |-  ( z  =  (/)  ->  ( y  e.  z  <->  y  e.  (/) ) )
75, 6mtbiri 676 . . . . . . . 8  |-  ( z  =  (/)  ->  -.  y  e.  z )
84, 7syl 14 . . . . . . 7  |-  ( z  e.  { x  e. 
{ (/) }  |  ph }  ->  -.  y  e.  z )
98adantl 277 . . . . . 6  |-  ( ( y  e.  z  /\  z  e.  { x  e.  { (/) }  |  ph } )  ->  -.  y  e.  z )
101, 9pm2.21dd 621 . . . . 5  |-  ( ( y  e.  z  /\  z  e.  { x  e.  { (/) }  |  ph } )  ->  y  e.  { x  e.  { (/)
}  |  ph }
)
1110gen2 1460 . . . 4  |-  A. y A. z ( ( y  e.  z  /\  z  e.  { x  e.  { (/)
}  |  ph }
)  ->  y  e.  { x  e.  { (/) }  |  ph } )
12 dftr2 4115 . . . 4  |-  ( Tr 
{ x  e.  { (/)
}  |  ph }  <->  A. y A. z ( ( y  e.  z  /\  z  e.  {
x  e.  { (/) }  |  ph } )  ->  y  e.  {
x  e.  { (/) }  |  ph } ) )
1311, 12mpbir 146 . . 3  |-  Tr  {
x  e.  { (/) }  |  ph }
14 ssrab2 3252 . . 3  |-  { x  e.  { (/) }  |  ph }  C_  { (/) }
15 ord0 4403 . . . . 5  |-  Ord  (/)
16 ordsucim 4511 . . . . 5  |-  ( Ord  (/)  ->  Ord  suc  (/) )
1715, 16ax-mp 5 . . . 4  |-  Ord  suc  (/)
18 suc0 4423 . . . . 5  |-  suc  (/)  =  { (/)
}
19 ordeq 4384 . . . . 5  |-  ( suc  (/)  =  { (/) }  ->  ( Ord  suc  (/)  <->  Ord  { (/) } ) )
2018, 19ax-mp 5 . . . 4  |-  ( Ord 
suc  (/)  <->  Ord  { (/) } )
2117, 20mpbi 145 . . 3  |-  Ord  { (/)
}
22 trssord 4392 . . 3  |-  ( ( Tr  { x  e. 
{ (/) }  |  ph }  /\  { x  e. 
{ (/) }  |  ph }  C_  { (/) }  /\  Ord  { (/) } )  ->  Ord  { x  e.  { (/)
}  |  ph }
)
2313, 14, 21, 22mp3an 1347 . 2  |-  Ord  {
x  e.  { (/) }  |  ph }
24 p0ex 4200 . . . 4  |-  { (/) }  e.  _V
2524rabex 4159 . . 3  |-  { x  e.  { (/) }  |  ph }  e.  _V
2625elon 4386 . 2  |-  ( { x  e.  { (/) }  |  ph }  e.  On 
<->  Ord  { x  e. 
{ (/) }  |  ph } )
2723, 26mpbir 146 1  |-  { x  e.  { (/) }  |  ph }  e.  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1361    = wceq 1363    e. wcel 2158   {crab 2469    C_ wss 3141   (/)c0 3434   {csn 3604   Tr wtr 4113   Ord word 4374   Oncon0 4375   suc csuc 4377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380  df-suc 4383
This theorem is referenced by:  ordtriexmid  4532  ontriexmidim  4533  ordtri2orexmid  4534  ontr2exmid  4536  onsucsssucexmid  4538  ordsoexmid  4573  0elsucexmid  4576  ordpwsucexmid  4581  unfiexmid  6930  exmidonfinlem  7205
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