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Theorem onintexmid 4323
Description: If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)
Hypothesis
Ref Expression
onintexmid.onint  |-  ( ( y  C_  On  /\  E. x  x  e.  y
)  ->  |^| y  e.  y )
Assertion
Ref Expression
onintexmid  |-  ( ph  \/  -.  ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem onintexmid
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prssi 3551 . . . . . 6  |-  ( ( u  e.  On  /\  v  e.  On )  ->  { u ,  v }  C_  On )
2 prmg 3519 . . . . . . 7  |-  ( u  e.  On  ->  E. x  x  e.  { u ,  v } )
32adantr 270 . . . . . 6  |-  ( ( u  e.  On  /\  v  e.  On )  ->  E. x  x  e. 
{ u ,  v } )
4 zfpair2 3973 . . . . . . 7  |-  { u ,  v }  e.  _V
5 sseq1 3021 . . . . . . . . 9  |-  ( y  =  { u ,  v }  ->  (
y  C_  On  <->  { u ,  v }  C_  On ) )
6 eleq2 2143 . . . . . . . . . 10  |-  ( y  =  { u ,  v }  ->  (
x  e.  y  <->  x  e.  { u ,  v } ) )
76exbidv 1747 . . . . . . . . 9  |-  ( y  =  { u ,  v }  ->  ( E. x  x  e.  y 
<->  E. x  x  e. 
{ u ,  v } ) )
85, 7anbi12d 457 . . . . . . . 8  |-  ( y  =  { u ,  v }  ->  (
( y  C_  On  /\ 
E. x  x  e.  y )  <->  ( {
u ,  v } 
C_  On  /\  E. x  x  e.  { u ,  v } ) ) )
9 inteq 3647 . . . . . . . . 9  |-  ( y  =  { u ,  v }  ->  |^| y  =  |^| { u ,  v } )
10 id 19 . . . . . . . . 9  |-  ( y  =  { u ,  v }  ->  y  =  { u ,  v } )
119, 10eleq12d 2150 . . . . . . . 8  |-  ( y  =  { u ,  v }  ->  ( |^| y  e.  y  <->  |^|
{ u ,  v }  e.  { u ,  v } ) )
128, 11imbi12d 232 . . . . . . 7  |-  ( y  =  { u ,  v }  ->  (
( ( y  C_  On  /\  E. x  x  e.  y )  ->  |^| y  e.  y
)  <->  ( ( { u ,  v } 
C_  On  /\  E. x  x  e.  { u ,  v } )  ->  |^| { u ,  v }  e.  {
u ,  v } ) ) )
13 onintexmid.onint . . . . . . 7  |-  ( ( y  C_  On  /\  E. x  x  e.  y
)  ->  |^| y  e.  y )
144, 12, 13vtocl 2654 . . . . . 6  |-  ( ( { u ,  v }  C_  On  /\  E. x  x  e.  { u ,  v } )  ->  |^| { u ,  v }  e.  {
u ,  v } )
151, 3, 14syl2anc 403 . . . . 5  |-  ( ( u  e.  On  /\  v  e.  On )  ->  |^| { u ,  v }  e.  {
u ,  v } )
16 elpri 3429 . . . . 5  |-  ( |^| { u ,  v }  e.  { u ,  v }  ->  ( |^| { u ,  v }  =  u  \/ 
|^| { u ,  v }  =  v ) )
1715, 16syl 14 . . . 4  |-  ( ( u  e.  On  /\  v  e.  On )  ->  ( |^| { u ,  v }  =  u  \/  |^| { u ,  v }  =  v ) )
18 incom 3165 . . . . . . 7  |-  ( v  i^i  u )  =  ( u  i^i  v
)
1918eqeq1i 2089 . . . . . 6  |-  ( ( v  i^i  u )  =  u  <->  ( u  i^i  v )  =  u )
20 dfss1 3177 . . . . . 6  |-  ( u 
C_  v  <->  ( v  i^i  u )  =  u )
21 vex 2605 . . . . . . . 8  |-  u  e. 
_V
22 vex 2605 . . . . . . . 8  |-  v  e. 
_V
2321, 22intpr 3676 . . . . . . 7  |-  |^| { u ,  v }  =  ( u  i^i  v
)
2423eqeq1i 2089 . . . . . 6  |-  ( |^| { u ,  v }  =  u  <->  ( u  i^i  v )  =  u )
2519, 20, 243bitr4ri 211 . . . . 5  |-  ( |^| { u ,  v }  =  u  <->  u  C_  v
)
2623eqeq1i 2089 . . . . . 6  |-  ( |^| { u ,  v }  =  v  <->  ( u  i^i  v )  =  v )
27 dfss1 3177 . . . . . 6  |-  ( v 
C_  u  <->  ( u  i^i  v )  =  v )
2826, 27bitr4i 185 . . . . 5  |-  ( |^| { u ,  v }  =  v  <->  v  C_  u )
2925, 28orbi12i 714 . . . 4  |-  ( (
|^| { u ,  v }  =  u  \/ 
|^| { u ,  v }  =  v )  <-> 
( u  C_  v  \/  v  C_  u ) )
3017, 29sylib 120 . . 3  |-  ( ( u  e.  On  /\  v  e.  On )  ->  ( u  C_  v  \/  v  C_  u ) )
3130rgen2a 2418 . 2  |-  A. u  e.  On  A. v  e.  On  ( u  C_  v  \/  v  C_  u )
3231ordtri2or2exmid 4322 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 662    = wceq 1285   E.wex 1422    e. wcel 1434    i^i cin 2973    C_ wss 2974   {cpr 3407   |^|cint 3644   Oncon0 4126
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-tr 3884  df-iord 4129  df-on 4131  df-suc 4134
This theorem is referenced by: (None)
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