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Theorem onintexmid 4378
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 3590 . . . . . 6  |-  ( ( u  e.  On  /\  v  e.  On )  ->  { u ,  v }  C_  On )
2 prmg 3556 . . . . . . 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 4028 . . . . . . 7  |-  { u ,  v }  e.  _V
5 sseq1 3045 . . . . . . . . 9  |-  ( y  =  { u ,  v }  ->  (
y  C_  On  <->  { u ,  v }  C_  On ) )
6 eleq2 2151 . . . . . . . . . 10  |-  ( y  =  { u ,  v }  ->  (
x  e.  y  <->  x  e.  { u ,  v } ) )
76exbidv 1753 . . . . . . . . 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 3686 . . . . . . . . 9  |-  ( y  =  { u ,  v }  ->  |^| y  =  |^| { u ,  v } )
10 id 19 . . . . . . . . 9  |-  ( y  =  { u ,  v }  ->  y  =  { u ,  v } )
119, 10eleq12d 2158 . . . . . . . 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 2673 . . . . . 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 3464 . . . . 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 3190 . . . . . . 7  |-  ( v  i^i  u )  =  ( u  i^i  v
)
1918eqeq1i 2095 . . . . . 6  |-  ( ( v  i^i  u )  =  u  <->  ( u  i^i  v )  =  u )
20 dfss1 3202 . . . . . 6  |-  ( u 
C_  v  <->  ( v  i^i  u )  =  u )
21 vex 2622 . . . . . . . 8  |-  u  e. 
_V
22 vex 2622 . . . . . . . 8  |-  v  e. 
_V
2321, 22intpr 3715 . . . . . . 7  |-  |^| { u ,  v }  =  ( u  i^i  v
)
2423eqeq1i 2095 . . . . . 6  |-  ( |^| { u ,  v }  =  u  <->  ( u  i^i  v )  =  u )
2519, 20, 243bitr4ri 211 . . . . 5  |-  ( |^| { u ,  v }  =  u  <->  u  C_  v
)
2623eqeq1i 2095 . . . . . 6  |-  ( |^| { u ,  v }  =  v  <->  ( u  i^i  v )  =  v )
27 dfss1 3202 . . . . . 6  |-  ( v 
C_  u  <->  ( u  i^i  v )  =  v )
2826, 27bitr4i 185 . . . . 5  |-  ( |^| { u ,  v }  =  v  <->  v  C_  u )
2925, 28orbi12i 716 . . . 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 2429 . 2  |-  A. u  e.  On  A. v  e.  On  ( u  C_  v  \/  v  C_  u )
3231ordtri2or2exmid 4377 1  |-  ( ph  \/  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 664    = wceq 1289   E.wex 1426    e. wcel 1438    i^i cin 2996    C_ wss 2997   {cpr 3442   |^|cint 3683   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-13 1449  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  ax-pr 4027  ax-un 4251  ax-setind 4343
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-ne 2256  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-int 3684  df-tr 3929  df-iord 4184  df-on 4186  df-suc 4189
This theorem is referenced by: (None)
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