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Theorem onsucelsucexmidlem 4439
Description: Lemma for onsucelsucexmid 4440. The set  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } appears as  A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5758), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4430. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
onsucelsucexmidlem  |-  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  e.  On
Distinct variable group:    ph, x

Proof of Theorem onsucelsucexmidlem
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 518 . . . . . . . 8  |-  ( ( ( y  e.  z  /\  z  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )  /\  z  =  (/) )  ->  y  e.  z )
2 noel 3362 . . . . . . . . . 10  |-  -.  y  e.  (/)
3 eleq2 2201 . . . . . . . . . 10  |-  ( z  =  (/)  ->  ( y  e.  z  <->  y  e.  (/) ) )
42, 3mtbiri 664 . . . . . . . . 9  |-  ( z  =  (/)  ->  -.  y  e.  z )
54adantl 275 . . . . . . . 8  |-  ( ( ( y  e.  z  /\  z  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )  /\  z  =  (/) )  ->  -.  y  e.  z )
61, 5pm2.21dd 609 . . . . . . 7  |-  ( ( ( y  e.  z  /\  z  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )  /\  z  =  (/) )  ->  y  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )
76ex 114 . . . . . 6  |-  ( ( y  e.  z  /\  z  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )  ->  (
z  =  (/)  ->  y  e.  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) } ) )
8 eleq2 2201 . . . . . . . . . . 11  |-  ( z  =  { (/) }  ->  ( y  e.  z  <->  y  e.  {
(/) } ) )
98biimpac 296 . . . . . . . . . 10  |-  ( ( y  e.  z  /\  z  =  { (/) } )  ->  y  e.  { (/)
} )
10 velsn 3539 . . . . . . . . . 10  |-  ( y  e.  { (/) }  <->  y  =  (/) )
119, 10sylib 121 . . . . . . . . 9  |-  ( ( y  e.  z  /\  z  =  { (/) } )  ->  y  =  (/) )
12 onsucelsucexmidlem1 4438 . . . . . . . . 9  |-  (/)  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }
1311, 12eqeltrdi 2228 . . . . . . . 8  |-  ( ( y  e.  z  /\  z  =  { (/) } )  ->  y  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )
1413ex 114 . . . . . . 7  |-  ( y  e.  z  ->  (
z  =  { (/) }  ->  y  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } ) )
1514adantr 274 . . . . . 6  |-  ( ( y  e.  z  /\  z  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )  ->  (
z  =  { (/) }  ->  y  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } ) )
16 elrabi 2832 . . . . . . . 8  |-  ( z  e.  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  ->  z  e.  { (/)
,  { (/) } }
)
17 vex 2684 . . . . . . . . 9  |-  z  e. 
_V
1817elpr 3543 . . . . . . . 8  |-  ( z  e.  { (/) ,  { (/)
} }  <->  ( z  =  (/)  \/  z  =  { (/) } ) )
1916, 18sylib 121 . . . . . . 7  |-  ( z  e.  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  ->  ( z  =  (/)  \/  z  =  { (/)
} ) )
2019adantl 275 . . . . . 6  |-  ( ( y  e.  z  /\  z  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )  ->  (
z  =  (/)  \/  z  =  { (/) } ) )
217, 15, 20mpjaod 707 . . . . 5  |-  ( ( y  e.  z  /\  z  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )  ->  y  e.  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )
2221gen2 1426 . . . 4  |-  A. y A. z ( ( y  e.  z  /\  z  e.  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )  ->  y  e.  { x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )
23 dftr2 4023 . . . 4  |-  ( Tr 
{ x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  <->  A. y A. z ( ( y  e.  z  /\  z  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )  ->  y  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } ) )
2422, 23mpbir 145 . . 3  |-  Tr  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }
25 ssrab2 3177 . . 3  |-  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  C_  { (/) ,  { (/)
} }
26 2ordpr 4434 . . 3  |-  Ord  { (/)
,  { (/) } }
27 trssord 4297 . . 3  |-  ( ( Tr  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  /\  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  C_  { (/) ,  { (/)
} }  /\  Ord  {
(/) ,  { (/) } }
)  ->  Ord  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )
2824, 25, 26, 27mp3an 1315 . 2  |-  Ord  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }
29 pp0ex 4108 . . . 4  |-  { (/) ,  { (/) } }  e.  _V
3029rabex 4067 . . 3  |-  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  e.  _V
3130elon 4291 . 2  |-  ( { x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }  e.  On 
<->  Ord  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )
3228, 31mpbir 145 1  |-  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  e.  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    \/ wo 697   A.wal 1329    = wceq 1331    e. wcel 1480   {crab 2418    C_ wss 3066   (/)c0 3358   {csn 3522   {cpr 3523   Tr wtr 4021   Ord word 4279   Oncon0 4280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285  df-suc 4288
This theorem is referenced by:  onsucelsucexmid  4440  acexmidlemcase  5762  acexmidlemv  5765
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