ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  acexmidlema Unicode version

Theorem acexmidlema 5731
Description: Lemma for acexmid 5739. (Contributed by Jim Kingdon, 6-Aug-2019.)
Hypotheses
Ref Expression
acexmidlem.a  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }
acexmidlem.b  |-  B  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  { (/) }  \/  ph ) }
acexmidlem.c  |-  C  =  { A ,  B }
Assertion
Ref Expression
acexmidlema  |-  ( {
(/) }  e.  A  ->  ph )
Distinct variable groups:    x, A    x, B    x, C    ph, x

Proof of Theorem acexmidlema
StepHypRef Expression
1 acexmidlem.a . . . 4  |-  A  =  { x  e.  { (/)
,  { (/) } }  |  ( x  =  (/)  \/  ph ) }
21eleq2i 2182 . . 3  |-  ( {
(/) }  e.  A  <->  {
(/) }  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } )
3 p0ex 4080 . . . . 5  |-  { (/) }  e.  _V
43prid2 3598 . . . 4  |-  { (/) }  e.  { (/) ,  { (/)
} }
5 eqeq1 2122 . . . . . 6  |-  ( x  =  { (/) }  ->  ( x  =  (/)  <->  { (/) }  =  (/) ) )
65orbi1d 763 . . . . 5  |-  ( x  =  { (/) }  ->  ( ( x  =  (/)  \/ 
ph )  <->  ( { (/)
}  =  (/)  \/  ph ) ) )
76elrab3 2812 . . . 4  |-  ( {
(/) }  e.  { (/) ,  { (/) } }  ->  ( { (/) }  e.  {
x  e.  { (/) ,  { (/) } }  | 
( x  =  (/)  \/ 
ph ) }  <->  ( { (/)
}  =  (/)  \/  ph ) ) )
84, 7ax-mp 5 . . 3  |-  ( {
(/) }  e.  { x  e.  { (/) ,  { (/) } }  |  ( x  =  (/)  \/  ph ) } 
<->  ( { (/) }  =  (/) 
\/  ph ) )
92, 8bitri 183 . 2  |-  ( {
(/) }  e.  A  <->  ( { (/) }  =  (/)  \/ 
ph ) )
10 noel 3335 . . . 4  |-  -.  (/)  e.  (/)
11 0ex 4023 . . . . . 6  |-  (/)  e.  _V
1211snid 3524 . . . . 5  |-  (/)  e.  { (/)
}
13 eleq2 2179 . . . . 5  |-  ( {
(/) }  =  (/)  ->  ( (/) 
e.  { (/) }  <->  (/)  e.  (/) ) )
1412, 13mpbii 147 . . . 4  |-  ( {
(/) }  =  (/)  ->  (/)  e.  (/) )
1510, 14mto 634 . . 3  |-  -.  { (/)
}  =  (/)
16 orel1 697 . . 3  |-  ( -. 
{ (/) }  =  (/)  ->  ( ( { (/) }  =  (/)  \/  ph )  ->  ph ) )
1715, 16ax-mp 5 . 2  |-  ( ( { (/) }  =  (/)  \/ 
ph )  ->  ph )
189, 17sylbi 120 1  |-  ( {
(/) }  e.  A  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    \/ wo 680    = wceq 1314    e. wcel 1463   {crab 2395   (/)c0 3331   {csn 3495   {cpr 3496
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502
This theorem is referenced by:  acexmidlem1  5736
  Copyright terms: Public domain W3C validator