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Theorem onsucelsucexmidlem 4282
Description: Lemma for onsucelsucexmid 4283. The set  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } appears as  A in the proof of Theorem 1.3 in [Bauer] p. 483 (see acexmidlema 5531), and similar sets also appear in other proofs that various propositions imply excluded middle, for example in ordtriexmidlem 4273. (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 489 . . . . . . . 8  |-  ( ( ( y  e.  z  /\  z  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )  /\  z  =  (/) )  ->  y  e.  z )
2 noel 3256 . . . . . . . . . 10  |-  -.  y  e.  (/)
3 eleq2 2117 . . . . . . . . . 10  |-  ( z  =  (/)  ->  ( y  e.  z  <->  y  e.  (/) ) )
42, 3mtbiri 610 . . . . . . . . 9  |-  ( z  =  (/)  ->  -.  y  e.  z )
54adantl 266 . . . . . . . 8  |-  ( ( ( y  e.  z  /\  z  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )  /\  z  =  (/) )  ->  -.  y  e.  z )
61, 5pm2.21dd 560 . . . . . . 7  |-  ( ( ( y  e.  z  /\  z  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )  /\  z  =  (/) )  ->  y  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )
76ex 112 . . . . . 6  |-  ( ( y  e.  z  /\  z  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )  ->  (
z  =  (/)  ->  y  e.  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) } ) )
8 eleq2 2117 . . . . . . . . . . 11  |-  ( z  =  { (/) }  ->  ( y  e.  z  <->  y  e.  {
(/) } ) )
98biimpac 286 . . . . . . . . . 10  |-  ( ( y  e.  z  /\  z  =  { (/) } )  ->  y  e.  { (/)
} )
10 velsn 3420 . . . . . . . . . 10  |-  ( y  e.  { (/) }  <->  y  =  (/) )
119, 10sylib 131 . . . . . . . . 9  |-  ( ( y  e.  z  /\  z  =  { (/) } )  ->  y  =  (/) )
12 onsucelsucexmidlem1 4281 . . . . . . . . 9  |-  (/)  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }
1311, 12syl6eqel 2144 . . . . . . . 8  |-  ( ( y  e.  z  /\  z  =  { (/) } )  ->  y  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )
1413ex 112 . . . . . . 7  |-  ( y  e.  z  ->  (
z  =  { (/) }  ->  y  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } ) )
1514adantr 265 . . . . . 6  |-  ( ( y  e.  z  /\  z  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )  ->  (
z  =  { (/) }  ->  y  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } ) )
16 elrabi 2718 . . . . . . . 8  |-  ( z  e.  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  ->  z  e.  { (/)
,  { (/) } }
)
17 vex 2577 . . . . . . . . 9  |-  z  e. 
_V
1817elpr 3424 . . . . . . . 8  |-  ( z  e.  { (/) ,  { (/)
} }  <->  ( z  =  (/)  \/  z  =  { (/) } ) )
1916, 18sylib 131 . . . . . . 7  |-  ( z  e.  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  ->  ( z  =  (/)  \/  z  =  { (/)
} ) )
2019adantl 266 . . . . . 6  |-  ( ( y  e.  z  /\  z  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )  ->  (
z  =  (/)  \/  z  =  { (/) } ) )
217, 15, 20mpjaod 648 . . . . 5  |-  ( ( y  e.  z  /\  z  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )  ->  y  e.  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )
2221gen2 1355 . . . 4  |-  A. y A. z ( ( y  e.  z  /\  z  e.  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )  ->  y  e.  { x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) } )
23 dftr2 3884 . . . 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 138 . . 3  |-  Tr  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }
25 ssrab2 3053 . . 3  |-  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  C_  { (/) ,  { (/)
} }
26 2ordpr 4277 . . 3  |-  Ord  { (/)
,  { (/) } }
27 trssord 4145 . . 3  |-  ( ( Tr  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  /\  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  C_  { (/) ,  { (/)
} }  /\  Ord  {
(/) ,  { (/) } }
)  ->  Ord  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )
2824, 25, 26, 27mp3an 1243 . 2  |-  Ord  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }
29 pp0ex 3968 . . . 4  |-  { (/) ,  { (/) } }  e.  _V
3029rabex 3929 . . 3  |-  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  e.  _V
3130elon 4139 . 2  |-  ( { x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }  e.  On 
<->  Ord  { x  e. 
{ (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )
3228, 31mpbir 138 1  |-  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) }  e.  On
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    \/ wo 639   A.wal 1257    = wceq 1259    e. wcel 1409   {crab 2327    C_ wss 2945   (/)c0 3252   {csn 3403   {cpr 3404   Tr wtr 3882   Ord word 4127   Oncon0 4128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133  df-suc 4136
This theorem is referenced by:  onsucelsucexmid  4283  acexmidlemcase  5535  acexmidlemv  5538
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