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Theorem onsucelsucexmidlem 4335
Description: Lemma for onsucelsucexmid 4336. The set  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } appears as  A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5625), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4326. (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 496 . . . . . . . 8  |-  ( ( ( y  e.  z  /\  z  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )  /\  z  =  (/) )  ->  y  e.  z )
2 noel 3288 . . . . . . . . . 10  |-  -.  y  e.  (/)
3 eleq2 2151 . . . . . . . . . 10  |-  ( z  =  (/)  ->  ( y  e.  z  <->  y  e.  (/) ) )
42, 3mtbiri 635 . . . . . . . . 9  |-  ( z  =  (/)  ->  -.  y  e.  z )
54adantl 271 . . . . . . . 8  |-  ( ( ( y  e.  z  /\  z  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )  /\  z  =  (/) )  ->  -.  y  e.  z )
61, 5pm2.21dd 585 . . . . . . 7  |-  ( ( ( y  e.  z  /\  z  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )  /\  z  =  (/) )  ->  y  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )
76ex 113 . . . . . 6  |-  ( ( y  e.  z  /\  z  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )  ->  (
z  =  (/)  ->  y  e.  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) } ) )
8 eleq2 2151 . . . . . . . . . . 11  |-  ( z  =  { (/) }  ->  ( y  e.  z  <->  y  e.  {
(/) } ) )
98biimpac 292 . . . . . . . . . 10  |-  ( ( y  e.  z  /\  z  =  { (/) } )  ->  y  e.  { (/)
} )
10 velsn 3458 . . . . . . . . . 10  |-  ( y  e.  { (/) }  <->  y  =  (/) )
119, 10sylib 120 . . . . . . . . 9  |-  ( ( y  e.  z  /\  z  =  { (/) } )  ->  y  =  (/) )
12 onsucelsucexmidlem1 4334 . . . . . . . . 9  |-  (/)  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }
1311, 12syl6eqel 2178 . . . . . . . 8  |-  ( ( y  e.  z  /\  z  =  { (/) } )  ->  y  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )
1413ex 113 . . . . . . 7  |-  ( y  e.  z  ->  (
z  =  { (/) }  ->  y  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } ) )
1514adantr 270 . . . . . 6  |-  ( ( y  e.  z  /\  z  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )  ->  (
z  =  { (/) }  ->  y  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } ) )
16 elrabi 2766 . . . . . . . 8  |-  ( z  e.  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  ->  z  e.  { (/)
,  { (/) } }
)
17 vex 2622 . . . . . . . . 9  |-  z  e. 
_V
1817elpr 3462 . . . . . . . 8  |-  ( z  e.  { (/) ,  { (/)
} }  <->  ( z  =  (/)  \/  z  =  { (/) } ) )
1916, 18sylib 120 . . . . . . 7  |-  ( z  e.  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  ->  ( z  =  (/)  \/  z  =  { (/)
} ) )
2019adantl 271 . . . . . 6  |-  ( ( y  e.  z  /\  z  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )  ->  (
z  =  (/)  \/  z  =  { (/) } ) )
217, 15, 20mpjaod 673 . . . . 5  |-  ( ( y  e.  z  /\  z  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )  ->  y  e.  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )
2221gen2 1384 . . . 4  |-  A. y A. z ( ( y  e.  z  /\  z  e.  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )  ->  y  e.  { x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )
23 dftr2 3930 . . . 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 144 . . 3  |-  Tr  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }
25 ssrab2 3104 . . 3  |-  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  C_  { (/) ,  { (/)
} }
26 2ordpr 4330 . . 3  |-  Ord  { (/)
,  { (/) } }
27 trssord 4198 . . 3  |-  ( ( Tr  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  /\  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  C_  { (/) ,  { (/)
} }  /\  Ord  {
(/) ,  { (/) } }
)  ->  Ord  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )
2824, 25, 26, 27mp3an 1273 . 2  |-  Ord  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }
29 pp0ex 4015 . . . 4  |-  { (/) ,  { (/) } }  e.  _V
3029rabex 3975 . . 3  |-  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  e.  _V
3130elon 4192 . 2  |-  ( { x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }  e.  On 
<->  Ord  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )
3228, 31mpbir 144 1  |-  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  e.  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 664   A.wal 1287    = wceq 1289    e. wcel 1438   {crab 2363    C_ wss 2997   (/)c0 3284   {csn 3441   {cpr 3442   Tr wtr 3928   Ord word 4180   Oncon0 4181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-tr 3929  df-iord 4184  df-on 4186  df-suc 4189
This theorem is referenced by:  onsucelsucexmid  4336  acexmidlemcase  5629  acexmidlemv  5632
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