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Theorem exmidontriimlem4 7160
Description: Lemma for exmidontriim 7161. 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 2221 . . 3  |-  ( b  =  B  ->  ( A  e.  b  <->  A  e.  B ) )
2 eqeq2 2167 . . 3  |-  ( b  =  B  ->  ( A  =  b  <->  A  =  B ) )
3 eleq1 2220 . . 3  |-  ( b  =  B  ->  (
b  e.  A  <->  B  e.  A ) )
41, 2, 33orbi123d 1293 . 2  |-  ( b  =  B  ->  (
( A  e.  b  \/  A  =  b  \/  b  e.  A
)  <->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) ) )
5 eleq2w 2219 . . . . . . 7  |-  ( b  =  w  ->  ( A  e.  b  <->  A  e.  w ) )
6 eqeq2 2167 . . . . . . 7  |-  ( b  =  w  ->  ( A  =  b  <->  A  =  w ) )
7 eleq1w 2218 . . . . . . 7  |-  ( b  =  w  ->  (
b  e.  A  <->  w  e.  A ) )
85, 6, 73orbi123d 1293 . . . . . 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 519 . . . . . . 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 520 . . . . . . . . . 10  |-  ( ( ( ( b  e.  On  /\  A. w  e.  b  ( ph  ->  ( A  e.  w  \/  A  =  w  \/  w  e.  A
) ) )  /\  ph )  /\  v  e.  b )  ->  ph )
18 eleq2w 2219 . . . . . . . . . . . . 13  |-  ( w  =  v  ->  ( A  e.  w  <->  A  e.  v ) )
19 eqeq2 2167 . . . . . . . . . . . . 13  |-  ( w  =  v  ->  ( A  =  w  <->  A  =  v ) )
20 eleq1w 2218 . . . . . . . . . . . . 13  |-  ( w  =  v  ->  (
w  e.  A  <->  v  e.  A ) )
2118, 19, 203orbi123d 1293 . . . . . . . . . . . 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 524 . . . . . . . . . . 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 2821 . . . . . . . . . 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 2530 . . . . . . . 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 2219 . . . . . . . . . 10  |-  ( v  =  y  ->  ( A  e.  v  <->  A  e.  y ) )
29 eqeq2 2167 . . . . . . . . . 10  |-  ( v  =  y  ->  ( A  =  v  <->  A  =  y ) )
30 eleq1w 2218 . . . . . . . . . 10  |-  ( v  =  y  ->  (
v  e.  A  <->  y  e.  A ) )
3128, 29, 303orbi123d 1293 . . . . . . . . 9  |-  ( v  =  y  ->  (
( A  e.  v  \/  A  =  v  \/  v  e.  A
)  <->  ( A  e.  y  \/  A  =  y  \/  y  e.  A ) ) )
3231cbvralv 2680 . . . . . . . 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 7159 . . . . . 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 4545 . . . 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 2530 . 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 2821 1  |-  ( ph  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ w3o 962    = wceq 1335    e. wcel 2128   A.wral 2435  EXMIDwem 4156   Oncon0 4324
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-setind 4497
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-uni 3774  df-tr 4064  df-exmid 4157  df-iord 4327  df-on 4329
This theorem is referenced by:  exmidontriim  7161
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