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Theorem lteupri 7604
Description: The difference from ltexpri 7600 is unique. (Contributed by Jim Kingdon, 7-Jul-2021.)
Assertion
Ref Expression
lteupri  |-  ( A 
<P  B  ->  E! x  e.  P.  ( A  +P.  x )  =  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem lteupri
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ltexpri 7600 . 2  |-  ( A 
<P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
2 ltrelpr 7492 . . . . 5  |-  <P  C_  ( P.  X.  P. )
32brel 4675 . . . 4  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
43simpld 112 . . 3  |-  ( A 
<P  B  ->  A  e. 
P. )
5 eqtr3 2197 . . . . . . . 8  |-  ( ( ( A  +P.  x
)  =  B  /\  ( A  +P.  y )  =  B )  -> 
( A  +P.  x
)  =  ( A  +P.  y ) )
6 addcanprg 7603 . . . . . . . 8  |-  ( ( A  e.  P.  /\  x  e.  P.  /\  y  e.  P. )  ->  (
( A  +P.  x
)  =  ( A  +P.  y )  ->  x  =  y )
)
75, 6syl5 32 . . . . . . 7  |-  ( ( A  e.  P.  /\  x  e.  P.  /\  y  e.  P. )  ->  (
( ( A  +P.  x )  =  B  /\  ( A  +P.  y )  =  B )  ->  x  =  y ) )
873expa 1203 . . . . . 6  |-  ( ( ( A  e.  P.  /\  x  e.  P. )  /\  y  e.  P. )  ->  ( ( ( A  +P.  x )  =  B  /\  ( A  +P.  y )  =  B )  ->  x  =  y ) )
98ralrimiva 2550 . . . . 5  |-  ( ( A  e.  P.  /\  x  e.  P. )  ->  A. y  e.  P.  ( ( ( A  +P.  x )  =  B  /\  ( A  +P.  y )  =  B )  ->  x  =  y ) )
109ralrimiva 2550 . . . 4  |-  ( A  e.  P.  ->  A. x  e.  P.  A. y  e. 
P.  ( ( ( A  +P.  x )  =  B  /\  ( A  +P.  y )  =  B )  ->  x  =  y ) )
11 oveq2 5877 . . . . . 6  |-  ( x  =  y  ->  ( A  +P.  x )  =  ( A  +P.  y
) )
1211eqeq1d 2186 . . . . 5  |-  ( x  =  y  ->  (
( A  +P.  x
)  =  B  <->  ( A  +P.  y )  =  B ) )
1312rmo4 2930 . . . 4  |-  ( E* x  e.  P.  ( A  +P.  x )  =  B  <->  A. x  e.  P.  A. y  e.  P.  (
( ( A  +P.  x )  =  B  /\  ( A  +P.  y )  =  B )  ->  x  =  y ) )
1410, 13sylibr 134 . . 3  |-  ( A  e.  P.  ->  E* x  e.  P.  ( A  +P.  x )  =  B )
154, 14syl 14 . 2  |-  ( A 
<P  B  ->  E* x  e.  P.  ( A  +P.  x )  =  B )
16 reu5 2689 . 2  |-  ( E! x  e.  P.  ( A  +P.  x )  =  B  <->  ( E. x  e.  P.  ( A  +P.  x )  =  B  /\  E* x  e. 
P.  ( A  +P.  x )  =  B ) )
171, 15, 16sylanbrc 417 1  |-  ( A 
<P  B  ->  E! x  e.  P.  ( A  +P.  x )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   E!wreu 2457   E*wrmo 2458   class class class wbr 4000  (class class class)co 5869   P.cnp 7278    +P. cpp 7280    <P cltp 7282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7291  df-pli 7292  df-mi 7293  df-lti 7294  df-plpq 7331  df-mpq 7332  df-enq 7334  df-nqqs 7335  df-plqqs 7336  df-mqqs 7337  df-1nqqs 7338  df-rq 7339  df-ltnqqs 7340  df-enq0 7411  df-nq0 7412  df-0nq0 7413  df-plq0 7414  df-mq0 7415  df-inp 7453  df-iplp 7455  df-iltp 7457
This theorem is referenced by:  srpospr  7770
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