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Theorem exmidontriimlem4 7201
Description: Lemma for exmidontriim 7202. The induction step for the induction on  A. (Contributed by Jim Kingdon, 10-Aug-2024.)
Hypotheses
Ref Expression
exmidontriimlem4.a  |-  ( ph  ->  A  e.  On )
exmidontriimlem4.b  |-  ( ph  ->  B  e.  On )
exmidontriimlem4.em  |-  ( ph  -> EXMID )
exmidontriimlem4.h  |-  ( ph  ->  A. z  e.  A  A. y  e.  On  ( z  e.  y  \/  z  =  y  \/  y  e.  z ) )
Assertion
Ref Expression
exmidontriimlem4  |-  ( ph  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A
) )
Distinct variable group:    y, A, z
Allowed substitution hints:    ph( y, z)    B( y, z)

Proof of Theorem exmidontriimlem4
Dummy variables  b  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2234 . . 3  |-  ( b  =  B  ->  ( A  e.  b  <->  A  e.  B ) )
2 eqeq2 2180 . . 3  |-  ( b  =  B  ->  ( A  =  b  <->  A  =  B ) )
3 eleq1 2233 . . 3  |-  ( b  =  B  ->  (
b  e.  A  <->  B  e.  A ) )
41, 2, 33orbi123d 1306 . 2  |-  ( b  =  B  ->  (
( A  e.  b  \/  A  =  b  \/  b  e.  A
)  <->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) ) )
5 eleq2w 2232 . . . . . . 7  |-  ( b  =  w  ->  ( A  e.  b  <->  A  e.  w ) )
6 eqeq2 2180 . . . . . . 7  |-  ( b  =  w  ->  ( A  =  b  <->  A  =  w ) )
7 eleq1w 2231 . . . . . . 7  |-  ( b  =  w  ->  (
b  e.  A  <->  w  e.  A ) )
85, 6, 73orbi123d 1306 . . . . . 6  |-  ( b  =  w  ->  (
( A  e.  b  \/  A  =  b  \/  b  e.  A
)  <->  ( A  e.  w  \/  A  =  w  \/  w  e.  A ) ) )
98imbi2d 229 . . . . 5  |-  ( b  =  w  ->  (
( ph  ->  ( A  e.  b  \/  A  =  b  \/  b  e.  A ) )  <->  ( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A
) ) ) )
10 exmidontriimlem4.a . . . . . . . 8  |-  ( ph  ->  A  e.  On )
1110adantl 275 . . . . . . 7  |-  ( ( ( b  e.  On  /\ 
A. w  e.  b  ( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A )
) )  /\  ph )  ->  A  e.  On )
12 simpll 524 . . . . . . 7  |-  ( ( ( b  e.  On  /\ 
A. w  e.  b  ( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A )
) )  /\  ph )  ->  b  e.  On )
13 exmidontriimlem4.em . . . . . . . 8  |-  ( ph  -> EXMID )
1413adantl 275 . . . . . . 7  |-  ( ( ( b  e.  On  /\ 
A. w  e.  b  ( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A )
) )  /\  ph )  -> EXMID )
15 exmidontriimlem4.h . . . . . . . 8  |-  ( ph  ->  A. z  e.  A  A. y  e.  On  ( z  e.  y  \/  z  =  y  \/  y  e.  z ) )
1615adantl 275 . . . . . . 7  |-  ( ( ( b  e.  On  /\ 
A. w  e.  b  ( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A )
) )  /\  ph )  ->  A. z  e.  A  A. y  e.  On  ( z  e.  y  \/  z  =  y  \/  y  e.  z ) )
17 simplr 525 . . . . . . . . . 10  |-  ( ( ( ( b  e.  On  /\  A. w  e.  b  ( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A
) ) )  /\  ph )  /\  v  e.  b )  ->  ph )
18 eleq2w 2232 . . . . . . . . . . . . 13  |-  ( w  =  v  ->  ( A  e.  w  <->  A  e.  v ) )
19 eqeq2 2180 . . . . . . . . . . . . 13  |-  ( w  =  v  ->  ( A  =  w  <->  A  =  v ) )
20 eleq1w 2231 . . . . . . . . . . . . 13  |-  ( w  =  v  ->  (
w  e.  A  <->  v  e.  A ) )
2118, 19, 203orbi123d 1306 . . . . . . . . . . . 12  |-  ( w  =  v  ->  (
( A  e.  w  \/  A  =  w  \/  w  e.  A
)  <->  ( A  e.  v  \/  A  =  v  \/  v  e.  A ) ) )
2221imbi2d 229 . . . . . . . . . . 11  |-  ( w  =  v  ->  (
( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A ) )  <->  ( ph  ->  ( A  e.  v  \/  A  =  v  \/  v  e.  A
) ) ) )
23 simpllr 529 . . . . . . . . . . 11  |-  ( ( ( ( b  e.  On  /\  A. w  e.  b  ( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A
) ) )  /\  ph )  /\  v  e.  b )  ->  A. w  e.  b  ( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A
) ) )
24 simpr 109 . . . . . . . . . . 11  |-  ( ( ( ( b  e.  On  /\  A. w  e.  b  ( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A
) ) )  /\  ph )  /\  v  e.  b )  ->  v  e.  b )
2522, 23, 24rspcdva 2839 . . . . . . . . . 10  |-  ( ( ( ( b  e.  On  /\  A. w  e.  b  ( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A
) ) )  /\  ph )  /\  v  e.  b )  ->  ( ph  ->  ( A  e.  v  \/  A  =  v  \/  v  e.  A ) ) )
2617, 25mpd 13 . . . . . . . . 9  |-  ( ( ( ( b  e.  On  /\  A. w  e.  b  ( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A
) ) )  /\  ph )  /\  v  e.  b )  ->  ( A  e.  v  \/  A  =  v  \/  v  e.  A )
)
2726ralrimiva 2543 . . . . . . . 8  |-  ( ( ( b  e.  On  /\ 
A. w  e.  b  ( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A )
) )  /\  ph )  ->  A. v  e.  b  ( A  e.  v  \/  A  =  v  \/  v  e.  A
) )
28 eleq2w 2232 . . . . . . . . . 10  |-  ( v  =  y  ->  ( A  e.  v  <->  A  e.  y ) )
29 eqeq2 2180 . . . . . . . . . 10  |-  ( v  =  y  ->  ( A  =  v  <->  A  =  y ) )
30 eleq1w 2231 . . . . . . . . . 10  |-  ( v  =  y  ->  (
v  e.  A  <->  y  e.  A ) )
3128, 29, 303orbi123d 1306 . . . . . . . . 9  |-  ( v  =  y  ->  (
( A  e.  v  \/  A  =  v  \/  v  e.  A
)  <->  ( A  e.  y  \/  A  =  y  \/  y  e.  A ) ) )
3231cbvralv 2696 . . . . . . . 8  |-  ( A. v  e.  b  ( A  e.  v  \/  A  =  v  \/  v  e.  A )  <->  A. y  e.  b  ( A  e.  y  \/  A  =  y  \/  y  e.  A ) )
3327, 32sylib 121 . . . . . . 7  |-  ( ( ( b  e.  On  /\ 
A. w  e.  b  ( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A )
) )  /\  ph )  ->  A. y  e.  b  ( A  e.  y  \/  A  =  y  \/  y  e.  A
) )
3411, 12, 14, 16, 33exmidontriimlem3 7200 . . . . . 6  |-  ( ( ( b  e.  On  /\ 
A. w  e.  b  ( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A )
) )  /\  ph )  ->  ( A  e.  b  \/  A  =  b  \/  b  e.  A ) )
3534exp31 362 . . . . 5  |-  ( b  e.  On  ->  ( A. w  e.  b 
( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A ) )  -> 
( ph  ->  ( A  e.  b  \/  A  =  b  \/  b  e.  A ) ) ) )
369, 35tfis2 4569 . . . 4  |-  ( b  e.  On  ->  ( ph  ->  ( A  e.  b  \/  A  =  b  \/  b  e.  A ) ) )
3736impcom 124 . . 3  |-  ( (
ph  /\  b  e.  On )  ->  ( A  e.  b  \/  A  =  b  \/  b  e.  A ) )
3837ralrimiva 2543 . 2  |-  ( ph  ->  A. b  e.  On  ( A  e.  b  \/  A  =  b  \/  b  e.  A
) )
39 exmidontriimlem4.b . 2  |-  ( ph  ->  B  e.  On )
404, 38, 39rspcdva 2839 1  |-  ( ph  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ w3o 972    = wceq 1348    e. wcel 2141   A.wral 2448  EXMIDwem 4180   Oncon0 4348
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-uni 3797  df-tr 4088  df-exmid 4181  df-iord 4351  df-on 4353
This theorem is referenced by:  exmidontriim  7202
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